Volume 20 Issue 2

MET

Medium Econometrische Toepassingen

The Erasmus Journal of Econometrics & Operations Research

On the other first Nobel laureate in Economics, and on confluence analysis An integrated risk estimation methodology: Ship specific incident type risk On an attempt to contribute to the safety of the shipping industry The ballot problem and the waiting time distribution in a non-preemptive queue with priorities

Edition 20-02

E M A S Marktinnovatie op het gebied van risicomanagement, integratie van financiële markten en veranderende IT-omgevingen zorgen ervoor dat het actuariële werkterrein voortdurend in beweging is. Actuarissen van vandaag begrijpen de ethische kwesties en psychologische factoren die invloed hebben op financieel risico en veiligheid. Hoe hebben deze kwesties invloed op het besluitvormingsproces? En, hoe heeft het nemen van passende actie invloed op het bedrijfsleven en de samenleving?

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2 Een opleiding die u alle vereiste kennis, vaardigheden en ondersteuning biedt om te slagen op het uitdagende terrein van risicomanagement.

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ACTUARIS: BESTE BEROEP 2010, 2011, 2012, 2013 EN 2014 (ELSEVIER/SEO)

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Preface Joris Blokland Erasmus University Rotterdam This edition of the MET (Medium Econometrische Toepassingen) is a very special one, because of multiple things. The first thing you’ll probably already have noticed, is the brand new design, which is designed by Marijn Waltman, member of the MET Committee.

We are very proud on this and we hope that you’ll enjoy it as much as we do. The new design is an outcome of the big change the Econometrisch Dispuut has made. No longer is the Faculty Association for Econometric & Operations Research Students named Econometrisch Dispuut, but FAECTOR! With this new name, comes a brand new design and so for the MET too.

Next to the changes in the design, there is also a new topic in the MET. We´ve asked Christiaan Heij, from the Econometric Institute, to write an profound article in the field of the Economics & Econometrics. His article about Nobel Prize Winners is very interesting and certainly worth reading it!

anen, is about the ballot problem. They derive a closed form expression of the waiting time distributions in an M/M/c queue with multiple priorities and a common service rate by using a combinatorial approach related to the well-known ballot problem.

With all the changes we’ve tried to make this MET as good as possible, because this is the last ‘regular’ printed MET magazine.

Due to growing prices of printing and the switch from printed magazines to online magazines, we have made the difficult decision to stop with making the printed version of the MET magazine. The MET will continue online on the new www.met-online.nl, where you can find all the previous editions and the new articles!

We wish you a lot of fun reading this MET!

The article of Sabine Knapp is about reducing risk in the shipping industry. Because of the growth of the shipping activity, the maritime administrations are trying to get a better understanding of total risk exposure. In her article she shows different insights and applications.

Jan Brinkhuis’ paper is also on the shipping industry, but a different aim. He does an attempt to contribute to the safety of the shipping industry. With a new ranking method, he tries to make the industry safer.

The article of van Vianen & Gabor, which is based on the Bachelor Thesis of Lars van Vi-

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Contents ■ Special On the other first Nobel laureate in Economics, and on confluence analysis Chirstiaan Heij ■ General An integrated risk estimation methodology: Ship specific incident type risk Sabine Knapp ■ General On an attempt to contribute to the safety of the shipping industry Jan Brinkhuis

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■ Logistics The ballot problem and the waiting time distribution in a non-preemptive queue with priorities Lars van Vianen Adriana F. Gabor

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Colophon Medium Econometrische Toepassingen (MET) is the scientific journal of FAECTOR, the faculty association for econometric students of the Erasmus University Rotterdam. Website: www.met-online.nl Editorial Board: Matthijs Aantjes, Marijn Waltman and Joris Blokland | Design: Marijn Waltman | Printer: Nuance Print, +31 - (0)10 - 592 33 62 | Circulation: 600 copies

lands | met@faector.nl | Acquisition: Max Schotsman | +31 - (0)10 - 408 14 39 ©2014 - No portion of the content may be directly or indirectly copied, published, reproduced, modified, displayed, sold, transmitted, rewritten for publication or redistributed in any medium without permission of the editorial board.

Address Editorial Board | Erasmus University Rotterdam | Medium Econometrische Toepassingen | Room H1102 | P.O. Box 1738 | 3000 DR Rotterdam | The Nether| MET | Edition 20-2 | FAECTOR

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On the other first Nobel laureate in Economics, and on confluence analysis Christiaan Heij Econometric Institute, Erasmus University Rotterdam Introduction Can two persons both be first-prize winners, without being ex aequo? Everyone, and especially every Rotterdam student in econometrics, knows that Jan Tinbergen was the first to win the Nobel Prize in Economics. In his brief speech at the Nobel Banquet in Stockholm, Tinbergen (1969) states: “I myself feel the honour bestowed on an econometrician as an honour bestowed on all the econometricians who from 1930 on have tried to develop this science or this branch of science. It is very difficult indeed to find out who did what. There is only one thing that I would like to add personally. There can be no doubt that Ragnar Frisch was the man who inspired us all in the annual meetings of the econometric society.”

See also Tinbergen (1974). And yes indeed, the Norwegian Ragnar Frisch was the other firstprize winner. As a pity, Frisch was not able to attend the festivities in Stockholm due to a broken leg, possibly related to his hobbies (Frisch, 1969): “My hobbies have been outdoor life, including mountain climbing on a modest scale.” It is also of interest to quote his motivation for choosing his field of study (Frisch, 1969, his italics): “My mother got a strong feeling that the trade would not be satisfactory for me in the long run. She insisted that at the same time as I completed my apprenticeship, I should take up a university study. We perused the catalogue of the Oslo University and found that economics was the shortest and easiest study.

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So, therefore, economics it became. That is the way it happened.”

Regression The name of Frisch is connected to the FrischWaugh (-Kloek-Lovell) theorem on the relation between multiple and partial regression. Another, and more influential, contribution was his socalled confluence analysis, and this is the topic of this note. We will use the conventional matrix notation of multiple regression, with observed data consisting of n observations of k+1 variables collected in the n×1 vectors (y, x1, ..., xk), with n (much) larger than k. The relation of interest is usually denoted by y = Xβ + ε, where X = (x1 ... xk) is the n×k regressor matrix, β is a k×1 vector of unknown parameters, and ε is an n×1 vector of unobserved errors. The ordinary least squares (OLS) estimates of β are obtained by minimizing the Euclidean norm ||y – Xb||2, with solution b = (X’X)-1X’y. Define ŷ = Xb, D̂ = (ŷ, X), and b̂ = (1, -b’)’, then it follows that D̂ b̂ = ŷ - Xb = 0. This means that, after replacing the variable y by ŷ, the k+1 variables (ŷ, x1, …, xk) are perfectly collinear. Further define y* = Xβ, D* = (y*, X), and β* = (1, -β’)’, then it follows that D*β* = y* - Xβ = 0. This provides the following interpretation of OLS. The unobserved (latent, error-free) variable y* satisfies the linear equation y* = Xβ exactly, whereas the observed (noisy) variable y = y* + ε does not. OLS determines the minimal possible distortion from y to ŷ, by minimizing ||y – ŷ||2, such that the approximating variable ŷ satisfies a linear relation ŷ = Xb perfectly.

The fundamental question raised by Frisch (1934) in his confluence analysis was why the variables

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Special are treated unequally in multiple regression, with a special role for the variable y. Such an asymmetric treatment may be reasonable if the purpose of modeling is (conditional) forecasting of y in terms of (x1, ..., xk), but in many cases one is simply interested in the (linear) relations among the k+1 observed variables, without any single variable standing out as being more important than the others. In confluence analysis, all variables are treated equally, and this raises the question in which way the variables should be approximated to obtain exact linear relations.

Total least squares Let D = (y, X) be the n×(k+1) matrix of observed data, then we seek for an error matrix E of the same dimensions such that D̂ = D - E satisfies one or more linear equations: D̂ c = 0 for some non-zero (k+1)×1 vector c. This condition means that D̂ should be rank deficient, and the number of (linearly independent) equations is equal to the dimension of the kernel of D̂ , that is, the nullity of D̂ . In terms of the (k+1)×(k+1) matrix of second-order moments M̂ = (1/n)D̂ ’D̂ , the main question is how to choose E and the associated nullity or corank of M̂ . The approximation error is defined in terms of the total sum of squares of all elements of the n×(k+1) error matrix E = D - D̂ , that is, the so-called (squared) Frobenius norm ||E||2 = ∑ ∑ eij2 = trace(E’E). Note that OLS is a special case, with D̂ = (ŷ, X), E = (y - ŷ, 0), and ||E||2 = ||y – ŷ||2.

Result 1 below shows how to approximate a given matrix by one of lower rank, in terms of the singular value decomposition (SVD). Let A be an n×m matrix of rank m, and let A’A have eigenvalues λ12 ≥ ... ≥ λm2 > 0, then there exist orthonormal matrices U of size n×n and V of size m×m such that U’U = In, V’V = Im, and A = UΛV’ where Λ = (Λ̃ 0)’ with Λ̃ the m×m diagnonal matrix with elements (λ1, …, λm) and with 0 the m×(n–m) zero matrix, so that Λ is n×m. This result can also be written as A = ∑jm=1 λjujv’j where uj and vj denote the j-th column of respectively U and V. As Avj = λjuj, a geometric interpretation is that A rotates the orthonormal basis (v1, ..., vm) to the m-dimensional subspace of ℝn spanned by the orthonormal vectors (u1, ..., um), where the j-th axis is re-scaled by the factor λj. The following reduced rank approximation of total least squares (TLS) is well-known in matrix algebra.

Result 1 (TLS) Let the SVD of the n×m matrix A be A = ∑jm=1 λjujv’j and let r < m, then Ar = ∑jr=1 λjujvj’ minimizes ||A – Ar|| under the restriction that rank(Ar) ≤ r, and ||A – Ar||2 = ∑jm=r+1 λj2. □ This TLS result can be applied for the n×(k+1) data matrix D, where we assume that rank(D) = k+1, that is, that the observed data are not perfectly collinear. Its SVD is denoted by D = UΛV’ and the reduced rank approximation by Dr. Let Vr consist of the last k+1–r columns of V; as V is orthonormal, v’jVr = 0 for all j = 1, ..., r, and hence DrVr = 0. Stated otherwise, the columns of the reduced rank matrix Dr satisfy k+1–r independent linear equations given by the columns of Vr. If r = k and λk+1 < λk, then the coefficient vector vk+1 of the equation Dkvk+1 = 0 is unique (up to a scaling constant).

Effect of data transformations Usually, the variables have no natural, self-evident measurement scales. OLS relations are invariant with respect to scale transformations of y and basis transformations on X. Indeed, let c ≠ 0 be a constant and let B be an invertible k×k matrix, then the OLS equation for the transformed data (yT , XT) = (cy, XB) obtained by regressing yT on XT is ŷT = XTbT with bT = (X’TXT)-1X’TyT = B-1(X’X)-1X’(cy) = cB-1b, so that XTbT = cb’X. The extension to confluence analysis, where all variables are treated equally, is to consider data transformations of the type DT = (yT , XT) = (y, X)T = DT where T is an invertible (k+1)×(k+1) matrix.

Approximating DT with respect to the Frobenius norm is equivalent to approximating the original data D with respect to the (squared) weighted 2 norm ||E||W = ||ET||2 = trace(T’E’ET) = trace(WE’E) where W = TT’. The optimal linear relation obtained by TLS (reducing the rank by 1 to r = k) is no longer invariant, as actually any relation can be obtained by suitable transformation.

Result 2 (Indeterminateness) If rank(D) = k+1, then for any non-zero (k+1)×1 vector a there exists an invertible (k+1)×(k+1) matrix T such that a is the coefficient vector of the TLS approximation of the transformed data DT; that is, the minimum of ||DT - DT,k|| under the restriction that rank(DT,k) ≤ k satisfies DT,ka = 0. □ The idea of the proof is rather simple, as Result 1 implies that it suffices to construct T such

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that the SVD of the transformed data is DT = ∑jk=+11 λj,Tuj,Tv’j,T with vk+1,T = a. To obtain T, define ã = a/||a|| and let (ṽ1, ..., ṽk) be such that Ṽ = (ṽ1, ..., ṽk, ã) is orthonormal. If the SVD of D is D = UΛV’, then take T = VṼ’ to get DT = DT = UΛV’VṼ’ = UΛṼ’, as desired. Result 2 shows the crucial importance of deciding on the choice of basis in the data space before approximating the observed data.

Errors in variables Result 2 shows that the TLS approximation depends completely on the chosen basis. Another disadvantage is that the approximation error E = D - Dr = ∑jk=+r1+1 λjujv’j is not truly “random”, as it satisfies the equations Evj = 0 for j = 1, ..., r.

To overcome these disadvantages, the usual assumption in confluence analysis is that each variable is observed under noise with independent noise components, the so-called errors-in-variables (EIV) assumption. Within this framework, the data are seen as a random sample of the k+1 variables (y, x1, ..., xk) where yi = y*i + ε0,i and xj,i = x*j,i + εj,i for i = 1, …, n, where the latent variables (y*i, x*1,i, ..., x*k,i) are perfectly collinear and the “noise” vector (ε0,i, ε1,i, ..., εk,i) is independent of the latent variables and has diagonal covariance matrix Mε. For simplicity, all variables are assumed to be demeaned so that both the latent variables and the noise terms have mean zero.

Let M* be the covariance matrix of the latent variables and M that of the observed variables, then the EIV assumptions imply that M = M* + Mε with Mε diagonal and M* rank-deficient, say M*a = 0 where a is a non-zero (k+1)×1 vector, and where M, M*, and Mε are symmetric positive semi-definite (PSD) (k+1)×(k+1) matrices. Because of the diagonal error covariance, the only allowed data transformations are DT = DT where T is an invertible diagonal matrix. The observed data second moment matrix M = (1/n)D’D is transformed to T’MT = T’M*T + T’MεT, where T’MεT is PSD diagonal and T’M*TT-1a = 0, so that the equation remains the same (the equation Da = 0 transforms to DTT-1a = 0, where DTT-1a = DTT-1a = Da).

For given observed data second moment matrix M, the decomposition M = M* + Mε is not unique and includes the k+1 regressions corresponding to cases where Mε has a single non-zero element.

Using the regression notation introduced before, let ŷ = Xb and define (y*i, x*1,i, ..., x*k,i) = (ŷi, x1,i, ..., xk,i) and (ε0,i, ε1,i, ..., εk,i) = (yi - ŷi, 0, ..., 0), then

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(*)

where we used that X’(y - ŷ) = X’y - X’Xb = 0 and ŷ’(y - ŷ) = b’X’(y - ŷ) = 0. Let D̂ = (ŷ, X), then M* = (1/n)D̂ ’D̂ is rank-deficient as M*a0 = 0 with a0 = (1, -b’)’. By choosing any of the k+1 variables as “dependent” variable, we get k+1 OLS relations M*aj = 0, for j = 0, ..., k. The following result is due to Klepper and Leamer (1984). Here it is assumed that the minimally achievable rank of M* is k, that is, the latent variables satisfy exactly one relation, and ℒ denotes the set of all (k+1)×1 coefficient vectors a such that M*a = 0. As each of these coefficient vectors is unique up to multiplication by a constant, they are scaled such that the first component is 1.

Result 3 (EIV) The k+1 OLS coefficient vectors a0, ..., ak are all in the same orthant if and only if M-1 is strictly positive (that is, with strictly positive elements), possibly after suitable choice of sign changes in the definition of the variables (±y, ±x1, ..., ±xk). In this case, ℒ consists of the convex hull of {a0, ..., ak}. □

The nice implication of this result is that, under the stated conditions, the sign of the coefficient of each specific variable is the same in all equations M*a = 0. So, although the equation itself is not unique, the sign of the coefficient of each of the k+1 variables is unique, and the set of all equations is given by a = ∑jk=0 cjaj where cj ≥ 0 and ∑jk=0 cj = 1.

As an illustration, Kalman (1982) considers the case k = 1, corresponding to two variables (y, x).

Denoting the second-order moments of the data by myy, mxx, and myx, the two OLS relations are (1, –myx/mxx)’ and (1, –myy/myx)’. Now consider any other relation a = (1, -α)’ such that M*a = 0.

Denote the second-order moments of the latent variables by m*yy, m*xx, and m*yx, then M = M* + Mε with Mε PSD diagonal implies that m*yy ≤ myy, m*xx ≤ mxx, and m*yx = myx. Further, M*a = 0 implies that myx - αm*xx = 0, so α = myx/m*xx, and det(M*) = 0 implies that m*yym*xx - my2x = 0, so that m*xx = my2x/m*yy ≥ my2x/myy. Therefore, my2x/myy ≤ m*xx ≤ mxx, and the value for α closest

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Special to 0 is myx/mxx whereas the value farthest away from 0 is myx/(my2x/myy) = myy/myx. The set of all solutions α consists of the interval between these two OLS values, and this shows Result 3 for k = 1.

Concluding remarks Hendry and Morgan (1989) explain why confluence analysis has virtually vanished from econometrics. One reason is the subjective choice of the noise structure (although OLS suffers from the same type of subjectivity), and another reason was the conversion to probabilistic reasoning starting with Haavelmo (1944). In some engineering areas, non-probabilistic TLS data modeling in line with Result 1 is very popular (Van Huffel and Vandewalle, 1987; Roorda, 1995). After standardizing each variable to unit variance, TLS (finding the singular vector corresponding to the smallest singular value of the data matrix D) is equivalent to finding the minimal eigenvalue λk2+1 and associated eigenvector of M = (1/n)D’D. The expression for M shown above in formula (*) shows that this eigenvector, say aTLS = (1, -b’TLS)’, satisfies (X’y, X’X - λk2+1I)aTLS = 0, so bTLS = (X’X - λk2+1I)-1X’y. This result shows that TLS can be seen as a kind of de-regularized OLS. Hendry and Morgan (1989) mention that the rank reduction idea of confluence analysis plays a role in cointegration analysis. Here the subjectivity due to the choice of the noise covariance matrix disappears because of differences in convergence speed. Suppose, for example, that each of the k+1 variables (y, x1, ..., xk) is integrated of order 1, so that the first difference of each variable is stationary (without deterministic trend), and that the variables satisfy a single cointegration relation so that a0y + ∑jk=1 ajxj is stationary (without deterministic trend). In this case, M = (1/n)D’D diverges for n → ∞, whereas a’Ma converges. Let M̃ = (1/n2)D’D, then M̃ converges to a matrix M∞ of rank k, and a’M∞a = 0 whereas b’M∞b ≠ 0 if b is not collinear with a. This means that, in the limit for n → ∞, the “latent” structure and the associated cointegration coefficients a are obtained from M∞. Testing for cointegration then amounts to testing whether the smallest eigenvalue of M∞ is zero. This clearly resembles confluence analysis, where the matrix of secondorder moments of the observed data is approximated by one of reduced rank. If the observed

variables are integrated whereas the noise is stationary, and if the moment matrix is appropriately scaled, then asymptotically the moment matrix of the noise vanishes and the moment matrix of the observed data converges to the moment matrix of the latent common trends. In this special case, the contribution of the noise disappears asymptotically, thereby eliminating identification problems that made confluence analysis less popular.

References

Frisch, R. (1934). Statistical Confluence Analysis by Means of Complete Regression Systems. University of Oslo Economic Institute, Publication no. 5.

Frisch, R. (1969). Biography. At http://www.nobelprize.org/nobel_prizes/economic-sciences/laureates/1969/frisch-bio.html (accessed March 10, 2014) Haavelmo, T. (1944). The probability approach in econometrics. Econometrica 12, Supplement, iii-vi and 1-115.

Hendry, D.F., and M.M. Morgan (1989). The re-analysis of confluence analysis. Oxford Economic Papers 41, 35-52.

Kalman, R.E. (1982). Identification from real data. In M. Hazewinkel and A.H.G. Rinooy Kan (eds.), Current Developments in the Interface: Economics, Econometrics, Mathematics. State of the Art Surveys Presented at the Occasion of the 25th Anniversary of the Econometric Institute, Erasmus University Rotterdam, January 1982, pp. 161-196. Reidel, Dordrecht.

Klepper, S., and E.E. Leamer (1984). Consistent sets of estimates for regressions with errors in all variables. Econometrica 52, 163-183.

Roorda, B. (1995). Global Total Least Squares. PhD thesis, Tinbergen Institute Research Series, volume 88, Erasmus University Rotterdam.

Tinbergen, J. (1969). Banquet speech. At http:// www.nobelprize.org/nobel_prizes/economic-sciences/laureates/1969/tinbergen-speech.html (accessed March 10, 2014) Tinbergen, J. (1974). Ragnar Frisch’s role in Econometrics – a sketch. European Economic Review 5, 3-6.

Van Huffel, S., and J. Vandewalle (1987). The Total Least Squares Problem: Computational Aspects and Analysis. Frontiers in Applied Mathematics, volume 9. Society for Industrial and Applied Mathematics, Philadelphia.

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An integrated risk estimation methodology: Ship specific incident type risk Sabine Knapp1 Econometric Institute, Erasmus University Rotterdam Abstract: Shipping activity has increased worldwide, including parts of Australia, and maritime administrations are trying to gain a better understanding of total risk exposure in order to mitigate risk. Total risk exposure integrates risk at the individual ship level, risk due to vessel traffic densities, physical environmental criteria, and environmental sensitivities. A comprehensive and robust risk exposure metric can be beneficial to maritime administrations to enhance mitigation of potential harm and reduce vulnerability to the marine environment as well as to safeguard lives and property. This report outlines an integrated methodology to estimate total risk exposure, with specific attention for the ship specific risk for different types of incident.

Some related application aspects of the models are discussed.

Keywords: Total risk exposure, binary logistic model, incident models, company risk estimation, visualization of risk dimensions 1. Introduction Most global trade is carried by sea, and shipping activity has increased by more than 300% since 1970 (UNCTAD, 2011). Growth in shipping activity increases the risk to marine ecosystems from pollution, shipping accidents, and spills. Figure 1 presents the framework for an integrated risk methodology. Total risk exposure integrates risk at the individual ship level, risk due to vessel traffic densities and composition, and physical environmental criteria. Risk exposure combined with sensitivities can be used to measure potential harm to property, life, or the marine environment, which can be reduced by risk control measures. A comprehensive and robust risk exposure metric can be beneficial to maritime administrations to enhance mitigation of potential harm and reduce vulnerability to the marine environment as well as to safeguard lives and property.

The Australian Maritime Safety Authority (AMSA) has refined a set of econometric models (Mueller, 2002, 2007, Knapp, 2011, 2006) that allow the estimation of various types and degrees of seriousness of risk, such as the probability of detention and incident types at individual ship and company level. The estimated models do not test for causality but rather identify relations between observed variables. This report provides a summary of the underlying methodology, building on Knapp (2006), to estimate incident risk, a crucial component of ship specific risk.. The data are presented in Section 2, the model outcomes in Section 3, and the interpretation of these outcomes in Section 4. Finally, Section 5 presents application examples of the incident type probabilities, such as company specific incident type risk and the combination and visualization of risk dimensions.

1 Corresponding author address: Econometric Institute, Erasmus School of Economics, Erasmus University Rotterdam, P.O. Box 1738, 3000 DR Rotterdam, Netherlands; knapp@ese.eur.nl. Disclaimer: The views expressed in this article represent those of the author and do not necessarily represent those of the Australian Maritime Safety Authority (AMSA).

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General 2. Underlying dataset and variables for incident type models The underlying sample data is a combination of ship particular data of the commercial world fleet, past inspection outcomes (number of deficiencies), and past ship incident data for the period January 2006 to December 2010. The data sources used for the analysis are IHS-Fairplay (IHSF), Lloyd’s Maritime Intelligence Unit (LMIU), the Global Integrated Ship Information System (GISIS) of the International Maritime Organization (IMO), and the Australian Maritime Safety Authority (AMSA). Data preparation for modeling is very important with respect to classification of incidents and the preparation of the dataset in general. Global incident information was combined from four different sources, and duplicates were eliminated. The remaining incidents were manually reclassified according to IMO definitions for seriousness which are very serious (including total loss), serious, and less serious incidents, defined as follows (IMO, 2000): - Very serious casualties (VS): are casualties to ships which involve total loss of the ship, loss of life, or severe pollution, the definition of which, as agreed by the Marine Environment Protection Committee at its thirty-seventh session (MEPC 37/22, paragraph 5.8), is as follows: ◦ Severe pollution: is a case of pollution which, as evaluated by the coastal State(s) affected or the flag Administration, as appropriate, produces a major deleterious effect upon the environment, or which would have produced such an effect without preventive action.

- Serious casualties (S): are casualties to ships which do not qualify as very serious casualties and which involve a fire, explosion, collision, grounding, contact, heavy weather damage, ice damage, hull cracking, or suspected hull defect, etc., resulting in: ◦ immobilization of main engines, extensive accommodation damage, severe structural damage, such as penetration of the hull under water, etc., rendering the ship unfit to proceed, or pollution (regardless of quantity); and/or ◦ a breakdown necessitating towage or shore assistance.

- Less serious casualties: are casualties to ships

which do not qualify as very serious casualties or serious casualties and for the purpose of recording useful information also include marine incidents which themselves include hazardous incidents and near misses.

AMSA’s incident data provides some near misses which were kept separate and used in lagged format with less serious incidents. Besides manual reclassification per seriousness, incident initial events were identified when possible which forms the basis of the models. This allows a better distinction between incident initial events and consequences. Missing data was whenever possible complemented to improve data quality.

The initial variables in the models and their respective groupings were selected based on Knapp and Franses (2007), Bijwaard and Knapp (2009) and Heij et al. (2011) but are extended due to the new and unique combination of data.

The groupings vary per incident type model.

Depending on the amount of observations, variables are grouped to facilitate implementation.

Due to the amount of variables with respect to the DoC2 company and beneficial ownership, the individual companies cannot be incorporated directly in the models as individual variables.

Their country of location was grouped using UNCTAD’s classification (UNCTAD, 2010) providing an indication of the level of development of a nation. These groups are developed nations, countries in transition, developing countries, and a category for unknown country of residence. The groups allow accounting partially for the effect of the company and provide a better basis to estimate risk at individual company level where the estimated probabilities are used. The explanatory variables included in the models are the following: • Ship type, age, and size (GRT) at the time of incident; • Classification society, flag; • Country where the vessel was built grouped into four groups as suggested by AMSA surveyors, and interaction effects with age groups (0-2 and above 14 years represent high age risk, while 3-14 years represent low age risk); • DoC company and group beneficial3 owner 2 Approximately 8000 companies could be identified by IMO company number.

3 Group beneficial ownership is defined by IHS-Fairplay.

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country of location classified according to country groups by UNCTAD; • Number of deficiencies within 360 days prior to the incident; • Number of incidents within 360 days prior to the incident; • Double hull indicator for tankers; • Changes of ship particulars overtime, such as flag changes, ownership changes, DoC company changes, class changes, and class withdrawals (within 3 years and within 5 years).

well-known logit model. This model states that

(1) where xi β is a weighted average of all explanatory factors:

Deficiency history information was aggregated and classified according to AMSA deficiency groups as follows: 1) Ship Certificates and Documents, 2) Human Factor – Crew Certificates, 3) Human Factor – Living and Working Conditions, 4) Human Factor – Operational, 5) ISM and Emergency Systems, 6) Life Saving Appliances, 7) Fire Fighting and Prevention, 8) Safety of Navigation and Communication, 9) Ship Structural and Machinery, 10) Pollution Prevention (split into noxious substances, air and all other.

3. Model combinations and model outcomes The models are estimated using historical data of the world fleet and global incident data for the time period January 2005 to December 2010.

A list of model types, dependent variables, and samples is given in Table 1. When possible, the incident type models are split up per seriousness and separate models are estimated.

The base model used to estimate the detention and incident type models is the binary logistic model. The end product is a set of formulas which can be used to estimate detention and incident probabilities at the individual ship level.

It has been demonstrated (Bijwaard and Knapp, 2009, Heij et al., 2011) that other models, such as duration analysis and the use of survival gains, provide alternative methods to quantify risk.

In all models considered here, the dependent variable (y) is binary, with two possible outcomes: “incident (1)” or “no incident (0)” Let xi contain the explanatory factors such as age, size, flag, classification society, and owner, then the logit model postulates that P(yi = 1|xi ) = F(xi β), where the weights β consist of a vector of unknown parameters and F is a cumulative distribution function (CDF). A popular choice is the CDF of the logistic distribution, which gives the

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The probabilities are estimated at the individual ship level (i), and the notation is explained in Table 2 (ℓ is the variable group counter, nℓ is the total number of classes within group ℓ, and k is an index from 1 to nℓ).

The coefficients are estimated by quasi-maximum likelihood (QML, Greene, 2000) to allow for possible misspecification of the assumed logistic CDF. Some summary statistics are presented in Appendix A. 4. Model results and interpretation The main purpose of the models is to fit probabilities and to estimate the effect of explanatory factors on these probabilities. For variables with positive (negative) coefficient, the risk increases (decreases) if the variable gets larger values. Categorical variables (e.g. flag, class, ship types) are compared to a benchmark, usually the class that is most common. The models are simplified by omitting insignificant variables (at the 5% or 1% significance level). Table 3 summarizes the main outcomes of the incident risk models, where the attention is restricted to the highest level of seriousness of incidents. For most incident types, age increases the risk (except for collisions, where younger vessels are more risk prone). Ship

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General size is significant for certain incident types and mostly increases the risk (except for main engine problems, where smaller vessels are more risk prone).

The effect of ship type is more often negative than positive, indicating that general cargo ships (the benchmark) are in most respects more risky than other ship types. This does not apply for incidents involving pollution and wrecked, stranded or groundings were no difference was found with respect to ship types. The variables indicating changes in ship particulars give mixed results. Class withdrawals increase the risk of incidents. Most combinations of where the vessel was built (country built) with age show a positive effect as compared to the benchmark (except for loss of life and equipment related incidents). Unknown country of location for DoC companies and ownership increases the risk. For developing nations and countries in transition, the results are mixed. Non IACS class or unknown class are not more risky than the benchmark (IACS class).

With respect to flag groups, black listed flags provide extra risk for about half of the models.

Lagged deficiency and incident history mainly show positive signs towards incident type risk.

Lagged very serious incidents and lagged serious incidents show a negative relationship with machinery related incidents and equipment related incidents (navigation/communication).

For most other incident types, lagged serious and less serious incidents show a positive sign towards further incident type risks with the exception of incident types fire and explosion and collisions.

Lagged deficiencies which have been evaluated with predictive value are deficiencies found in the area of ISM, crew certificates and qualifications, ship certificates and documentation, Living and Working Conditions, operational deficiencies, Fire Fighting (FFP) and prevention, Radio Communications and Life Saving Appliances (LSA).

Recall that the models do not test for causality.

The deficiency types are evaluated for predictive value.

5. Model applications examples and visualization of risk dimensions Besides the application of the incident type models to account for ship specific risk in the

estimation of total risk exposure (Figure 1), some other applications are presented here.

One of such applications is to use the incident type probabilities to estimate risk for individual DoC companies or beneficial ownership companies. The topic to estimate risk at individual DoC company level was treated by AMSA in a previous report by the CSIRO (Mueller (2007)) with restricted application to AMSA inspection data.

Heij and Knapp (2012) built on this methodology and present two other methods to estimate the risk of very serious and serious incidents.

The analysis of global fleet data and incident data revealed some weaknesses, in particular missing company data. This issue was raised recently at IMO Council, since it is connected to the evaluation of the management of the IMO numbering schemes (IMO, 2011). Due to the large amount of DoC companies and beneficial ownership, the companies cannot be evaluated individually in the models. Their country of location was grouped using UNCTADâ&#x20AC;&#x2122;s classification (UNCTAD, 2010) providing an indication of the level of development of a nation. The results are mixed, but the groups account partially for the effect of the company and provide a better basis to estimate risk at individual company level.

The underlying idea to estimate risk at individual company level is based on the following concept. Given the number of ships (N) under the management of a certain company, its number of observed incidents, and its model based mean probability of incident (p), tail probabilities are calculated by means of the binominal distribution. A company is risky if its actual number of incidents is higher than expected from the model probabilities, and the right-tail probability is the probability to observe the actual number of incidents or more given N trials with modelbased incident probability. If this right-tail probability is small, the company is riskyFor incident risk, 3785 companies could be evaluated. The method identified 80 companies (2.1%) as risky for the class of very serious incidents and 137 (3.6%) as risky for series incidents.

Another possible application of the model-based probabilities is to visualize risk dimensions â&#x20AC;&#x201C; that is to combine various risk types in two dimensions. For that we refer to the underlying methodology developed by Heij and Knapp (2012),

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which is briefly summarized here to demonstrate application aspects. Ship specific risk has two dimensions – preventive type of risk (detention) and incident type risk. The correlation between the two at individual ship level turned out to be relatively low. This means that ships with a high probability of detention do not necessarily show a high probability of incident. It has been demonstrated in the literature (Knapp, 2006, Bijwaard and Knapp, 2009) that inspections decrease incident type risk, hence a vessel benefits from an inspection.

Ship specific incident type risk can be expressed in terms of probabilities with a possible extension to estimate the monetary value at risk (MVR), a measure for consequences. MVR is a combination of the total insured value (TIV) of a vessel of five damage types and incident type probabilities (Knapp et all, 2011). The five damage types are 1) hull and machinery damage, 2) insured value of life, 3) oil pollution, 4) third party liability limits, and 5) cargo values and the corresponding incident type probabilities to combine with the TIV values are as follows: 1) Probability of damage to hull or machinery for all ships, 2) Probability of loss of life for passenger vessels, 3) Probability of pollution for oil tankers, 4) Probability of third party liability for all ships and 5) Probability of cargo damage for all ships except passenger vessels.

The calculation of MVR according to Heij and Knapp (2012) is given in equation 2 where pinc is the probability of an incident, pj is the conditional probability of damage type j in case of an accident, and Vj is the monetary value of this damage type. The conditional probability of damage type j and the values Vj are constructed in Knapp et al. (2011).

(2)

For the combination and visualization of risk dimensions, Heij and Knapp (2012) estimate three major components at individual ship level as follows: 1) probability of detention, 2) probability of five incident types and 3) the estimated

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monetary value at risk (MVR). If all components are known, the estimated monetary value at risk (MVR) at individual ship level can be calculated and graphically combined with the probability of detention.

A simple example is given in Figure 2 based on vessels that arrived during a three day period in the port of Newcastle in Australia. In this example, the ships are dry bulk carriers of a certain profile. Detention risk is plotted on the horizontal axis while MVR is plotted on the vertical axis.

Risk graduation is visualized by color from blue (low risk) to red (high risk). It visualizes in two dimensions how each ship relates to the full fleet in terms of overall risk. The plot can be accompanied with a set of numerical values which describe the location of each vessel.

Acknowledgements The author wishes to thank the Australian Maritime Safety Authority, IHS-Fairplay, and LMIU for providing the necessary data. The author also wishes to thank Dave Nagle from RightShip for his support provided in preparing the underlying data matrix and numerous data matching routines. The author further likes to acknowledge the contribution of co-author Christiaan Heij for the application examples presented in section 5.

References

Bijwaard G and Knapp S (2009), Analysis of Ship Life Cycles – The Impact of Economic Cycles and Ship Inspections, Marine Policy, 33(2): 350-369 Greene H.W (2000), Econometric Analysis, Fourth Edition, Prentice Hall, New Jersey Heij C, Bijwaard G and Knapp S (2011), Ship Inspection Strategies: Effects on Maritime Safety and Environmental Protection, Transportation Research Part D, 16, 42-48 Heij C, Knapp S (2012), Evaluation of safety and environmental risk at individual ship and company level, Transportation Research Part D 17 228–236 IMO (2000), MSC/Circ. 953, MEPC/Circ. 372, Reports on Marine Casualties and Incidents, Revised harmonized reporting procedures, adopted 14th December 2000, IMO, London.

IMO (2011), C 106/4/3/Rev.1, IMO Ship Identification Number scheme and IMO Unique Company and Registered Owner Identification Number scheme, 31st May 2011, IMO, London Knapp, S (2006), The Econometrics of Maritime Safety – Recommendations to improve safety at sea, Doctoral Thesis, Erasmus University Rotterdam Knapp S, (2011), Integrated risk estimation meth-

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General odology and implementation aspects (Main Report AMSA); Reference Nr. TRIM 2010/860, 2011-1a (Main Report AMSA) Knapp, S, Franses, P.H (2007), A global view on port state control - econometric analysis of the differences across port state control regimes. Maritime Policy and Management 34, 453-483.

Knapp S, Bijwaard G and Heij C (2011), Estimated Incident Cost Savings in Shipping due to Inspections, Accident Analysis and Prevention, 43: 1532â&#x20AC;&#x201C;1539 Mueller W, Morton R (2002), Statistical Analysis of AMSA Ship Inspection Data, 1995-2001, CSIRO Report Nr. CMIS 2002/01, Canberra, report provided in confidence Mueller W (2007), Statistical Modelling of Ship Detention based on Ship Inspection Data 1996 to 2005, CSIRO Report Nr. CMIS 07/128, Canberra, report provided in confidence UNCTAD, 2010, 2011, Review of Maritime Transport 2010, 2011, UNCTAD, Geneva

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On an attempt to contribute to the safety of the shipping industry Jan Brinkhuis Econometric Institute Rotterdam Introduction The need for a safer shipping industry. There is a compelling need to improve the safety of the shipping industry. Occurrence of very serious accidents and inspections of ships where so many shortcomings are discovered that detainment becomes necessary, are constant reminders of this need. At first sight, it might seem simple to achieve great improvements of this safety. Eliminate substandard ships from the seas and all will be well, you might think.

Effort flags not observable. It is the obligation of the flag under which a ship sails to make sure that the ship satisfies internationally agreed standards. It is known that some flags do not put enough effort into this task, to put it mildly.

So why not give bad flags the choice between improving their performance or face severe consequences of their underperformance? Unfortunately, things are not so simple and clear cut in the shipping industry. The main problem is here that it is not possible to monitor the effort that flags put into doing their job, which would give a base for taking action against underperforming flags.

Black list. The only information one can obtain about the effort of a flag is indirect: if its ships are involved in many very serious accidents and are often detained after inspections, then one can conclude that the effort is not adequate. This is not really hard evidence and so this information cannot be used for taking direct measures against flags where the numbers of very serious accidents and detainments are high. One can only do two things. First, one can shame such

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flags by putting these flags on a black list and publish this list. This is not good for the reputation for a flag and as a result these flags will lose business: good ships will move to flags with a better reputation. Second, each time a ship from a black list flag enters a port, one can target it for inspection; needless to say that this is very irritating for the ships involved. Other ships are only inspected randomly with a small probability. It is not hard to understand that the choice of method for composing the black list is a very sensitive issue.

Shortcomings of current ranking method. The method in current use to rank flags is considered to be not adequate. There are actually three lists: black, grey and white, each one made according to a different criterion. The method makes use of the excess factor, a technical device that is not really well understood by the parties involved (only by the experts that make the lists). Moreover, it is unfair. For example, some flags specialize in new ships, some in old ships. It is clear that the former have a much easier job. In the current system these tend to end up on the white list, but the latter tend to end up on the black list. Another injustice is that for flags with a small fleet a few accidents lead already to a place on the black list and so these flags are more susceptible to bad luck.

A new ranking method. The present paper is based on a very recent paper, `A method to measure enforcement effort in shipping with incomplete information' by Xichen Ji, Jan Brinkhuis, Sabine Knapp, Economic Institute Report 201412 (submitted to an international scientific jour-

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General nal); this contains more information than the present paper, and it contains references to other papers on the subject. It proposes a new method to rank flags that is based on one simple and convincing criterion (the number of undesirable events that are caused by inadequate effort) and then the worst quartile is considered high risk (proxy to low effort), the second worst medium risk (proxy to medium effort) and the best two quartiles low risk (proxy to high effort).

Advantages of the new method. This method is fair to all flags. Moreover, it makes use of a very important economic principle: it makes sure that the incentives to the parties involved are the right ones. As far as we can see, it solves all shortcomings of the system in current use. In particular, it takes into account the composition of the fleet of a flag (`old or new vessels') and it gives just enough sympathy for bad luck to make sure that flags do not get undeservedly a low ranking (`in particular, flags with a small fleet will not be punished for bad luck'). That we managed to find natural solutions to deal with shortcomings of the current method in the context of the new method, is proof how flexible this new method is.

It was an interesting challenge for the authors to come up with good models and adequate techniques to overcome all hurdles; in the present paper, we will explain how these problems were solved by adequate techniques. It is hoped that this new method will be implemented in practice and that it will contribute to a greater safety of the shipping industry.

Scope of the new method. The new method is not limited to the shipping industry. Readers are challenged to try to think of ways in which the method explained below could be applied for example in the context of health care.

1. Background Let us be more systematic!

PSC Inspections. The shipping industry is characterized by a complex legislative framework of over 50 conventions of the International Maritime Organization (IMO), which unfortunately lacks enforcement powers of due to its international nature. Enforcement of internationally agreed standards is the obligation of a registry or flag. It is not applied with equal force, and this creates opportunities for substandard shipping.

Since enforcement at the flag state level is not directly monitored, port states have created port state control regimes (PSC) that enforce internationally agreed standards on vessels entering their territory, by exercising their right to perform PSC inspections. If a vessel is found to be not compliant, it can be detained.

BGW List. Two PSC regimes (the Paris MoU and the Tokyo MoU) publish each year a list that rank flags according to their performance during inspections, the so called Black/Grey/White List (BGW-list), where black listed flags perform worst. The list has become the industry standard and is seen as a proxy to measure the effort (or quality) of enforcement of a registry, despite the shortcomings of this list. The list is also used by the Paris MoU and the Tokyo MoU to target ships for inspection.

IMO and improving the safety of the industry. Other relevant developments dealing with performance measurement are taking place at the IMO. For example, Assembly Resolution A.1037(27) (IMO, 2013) asks for the development of performance indicators that can measure progress made towards its broad strategic directions including improving the safety of the industry. One of these directions deals with fostering global compliance, but this is not measured at the individual member state level. However, the IMO has established the Member State Audit scheme, where individual audits are performed, primarily based on qualitative measures.

Shortcomings of the current method. There is the inability of the current method to deal with small sample sizes as well as with large numbers of inspections; for example, large numbers of inspections can lead to complications since these can make the grey list so narrow that it becomes incomparably smaller than the black and white ones. The list is also characterized by the lack of a satisfactory criterion for the effort of a flag.

Indeed, the criterion is defined in terms of the excess factor, the value of which depends on the BGW-list and for each of the three, black/grey/ white, it is defined by a different procedure. Performance is currently only measured based on inspection outcomes such as detentions. In principle, other factors might also be relevant, such as the age of the vessel, the sizes or the ship type, as these have an influence on the safety quality

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of ships. For ship types, the reason for this is that the major shipping markets have different characteristics. Moreover, it is also dicult to evaluate a flag with a small fleet fairly by means of currently used methods. One reason for this is that for small fleet, the performance is more prone to bad luck. In addition, the current method also suffers from the lack of coordination amongst PSC regimes to use combined data. In a paper

by Perepelkin et al. (2010) some of these problems have been addressed and solved; the new method that is proposed below builds on some aspects of their work.

2. A crude measure for the performance of a flag The idea: count undesirable events. We begin by explaining how the proposed method is based on a simple, clear and convincing idea. It is reasonable to expect that in case of sufficient effort by a flag, for the ships under this flag, certain undesirable events will be rare. For example, inspections of ships will rarely lead to detention, and very serious incidents will be rare. This suggests to count some well-chosen types of undesirable events, detentions and very serious accidents, and to use the outcome as a performance measurement that is proxy for the effort: a low respectively high outcome is interpreted as a good respectively inadequate effort by the flag. This simple idea is the base of the proposed method. Of course, one cannot just count the total number of all undesirable events for a flag, detentions and very serious accidents. In order to be fair, one has to take into consideration, for each flag, its total number of inspections, its fleet size and the fact that both types of undesirable events do not have equal weight.

Performance measure: crude. This leads to the introduction of two numbers for each flag F, dF, the quotient of the proportion of inspections of vessels under flag F that lead to detention and this proportion for all vessels, and zF, the quotient of the proportion of the vessels under flag F that has been involved in a very serious accident and this proportion for all vessels. Thus, we get that for dF, as well as for zF, the value 1 is a benchmark. For example, zF is smaller, respectively larger, than 1 precisely if the proportion of vessels under flag F that has been involved in a very serious accident is smaller, respectively

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larger, than this proportion for all flags. Then we can compare the effort of two flags, F1 and F2, for which dF1 â&#x2030;Ľ dF2 and zF1 â&#x2030;Ľ zF2: in this case, we consider that the effort of F2 is at least as good as that of F1. We want to extend this idea in order to be able to compare the effort for each pair of flags. To this end, we introduce a weight factor c, to be chosen by policy makers. Then we consider that the effort of F2 is at least as good as that of F1 precisely if dF1 + czF1 â&#x2030;Ľ dF2 + czF2 That is, we take as a first-crude-attempt the following formula for measuring the performance of a flag F: 'crude performance measure' (1) The lower this number is, the better the effort of the flag to enforce standards.

3. A finer measure for the performance of a flag taking ship types into account Ship types. In order to better quantify the effort of a flag, we feel that it is best to distinguish between ship types: 1) general cargo, 2) dry bulk carrier, 3) container vessel, 4) tankers, 5) passenger vessels and 6) all other ship types.

One reason for taking ship type into account is that ship type can be used as a proxy for the characteristics of the market a particular vessel trades in. These characteristics are determined by the nature of the trade flows, the legislative framework and the economic pressures and are relevant for measuring the performance of a flag. For example, tankers or some dry bulk carriers are subject to industry vetting inspections besides port state control inspections. Another example is that container vessels operate under regular liner trades and carry higher value cargo with better safety quality management. Other reasons for taking ship type into account are that ship types can also be used as a proxy for other factors such as age or size.

Performance measure: finer. Now we come to the second-finer-attempt to measure the performance of a flag. We make just one small change to the crude formula: we take ship types into account. This gives automatically the following improved formula for measuring the performance of a flag F as will be explained below:

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'finer performance measure' (2)

General where: • F: a flag, • QF: the performance measure of flag F; • NF: the number of inspections of ships under flag F during the period under consideration; • Nships F: the number of ships under flag F, averaged over the period under consideration; • c: a positive constant, to be chosen by policymakers, that gives the weight of a very serious casualty compared to a detention; • t: a type of ship, determined by age and tonnage group; • t ∈ F: shorthand notation for 'the types of ship that occur among the ships under flag F'; • Dt,F: the number of detentions of ships of type t under flag F during the period under consideration; • Zt,F: the number of very serious incidents of ships of type t under flag F during the period under consideration; The coefficients αt and βt in the formula above are calculated with the following formulas: • αt = Nt /Dt provided Dt is not zero, where Dt is the number of detentions of ships of type t of all flags during the period under consideration, and Nt is the number of inspections of ships of type t of all flags during the period under consideration; if Dt = 0, then we put αt = 0, for example (it does not matter what we put here, as in the summation t is multiplied with Dt,F, which is zero if Dt = 0.)

• βt = Nships t / Zt if Zt is not zero, where Nships t is the number of ships of type t of all flags during the period under consideration, and where Zt is the number of very serious incidents of ships of type t for all flags, during the period under consideration; if Zt = 0, then we put βt = 0, for example (with a similar justification as given above for αt).

Precise motivation for the finer performance measure. The reason for the chosen correction for detentions is as follows: without corrections for ship types, we would take for the contribution of the detentions to the measure of the performance of flag F, the ratio DF /NF, where DF is the number of detentions of ships under flag F. This can be written as

.

To make the numbers of detentions comparable between different types, it is reasonable to multiply, for each type t, the term Dt,F by αt = Nt / Dt, the average number of inspections for one detention for ships of type t.

This gives the contribution to the measure of the enforcement effort of flag F. In particular, this will make the contribution of the detentions of old ships smaller, as desired.

The reason for the chosen correction for very serious incidents is the same. In particular, for a flag that has an average number of detentions and very serious incidents, the enforcement effort will be 1 + c.

4. Final measure for performance of a flag, giving sympathy for bad luck Giving the benefit of the doubt. Our final measure for the performance of a flag, takes the difference in variations in the observations for small and large fleets into account. The finer measure of the performance of a flag given above, QF, is not an adequate measure for the performance of flags with a small fleet. The numbers Dt,F and Zt,F are subject to chance, they are stochasts, and for small fleet the effect of chance on the outcomes (for example, 'bad luck'), and so on the outcome of the formula, can be unacceptably large. Ideally we would like to replace in the formula the stochasts by their expectations, but these are unknown. Therefore, we will give a systematic method to replace the numbers Dt,F and Zt,F by numbers that are slightly smaller, just enough to make sure, within a certain precision, that these numbers are smaller than the expectations of these stochasts. This makes the outcome of the formula smaller, and so a flag is certain not to be given by bad luck a performance measure QF that is higher than deserved; to be more precise: it is very unlikely that this will happen. That is, the qualities of the flags are attributed with some sympathy, 'the benefit of the doubt'. We will prove below that this systematic way accords more sympathy to flags with a small fleet than to flags with a larger fleet, as desired.

Thus the shortcoming of the finer performance measure above is repaired.

Analysis of the variations in the observations.

Next we provide an analysis for the variation in the observations. So far, the formulas represent

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no more than the adding up of numbers. However the analysis we are now going to carry out requires more subtle calculations. We will try to make clear that the underlying idea is again simple: it is the wish to attribute some sympathy but not more than is needed. We introduce the following additional notation: Nt,F , the number of inspections of ships of type t under flag F during the period under consideration (Σt∈F Nt,F = NF), and Nships t,F , the total number of ships of type t under flag F, averaged over the period under consideration (Σt∈F Nships t,F = Nships F ). Assume that the number of detentions of a certain ship type under a certain flag follows a binomial distribution: Dt,F ~ Bin(Nt,F ,pdt,F), with pdt,F the underlying probability of detention at one inspection of ship type t under the flag F. Moreover, we assume that the number of very serious incidents of ship type t under flag F also follows a binomial z z distribution Zt,F ~ Bin(Nships t,F ,pt,F ), with pt,F the underlying probability of a very serious casualty for each ship of type t under flag F. The probabilities of detention or very serious casualties of one ship type differ among the flags because of the differences in management among the flags, the very effect we want to measure. When Nt,F is large enough, the distribution of the stochast Dt,F approximates the normal distribution with E(Dt,F) = Nt,F*pdt,F and Var(Dt,F) = Nt,F*pdt,F*(1-pdt,F).

The same holds for the distribution of stochast Zt,F: when Nships t,F is large enough, the distribution of the stochast Zt,F approximates a normal z distribution, with E(Zt,F) = Nships t,F*pt,F and Var(Zt,F) z z = Nships t,F*pt,F*(1-pt,F). It will be convenient to write (pdt,F)' = Dt,F /Nt,F and to view this as an observation of a normally distributed stochast with mean pdt,F and variation (pdt,F (1-pdt,F))/Nt,F. Similarly z for (pt,F )'= zt,F /Nships t,F. Ideally we would like to take as a measure for the performance of a flag:

'ideal final preformance measure' (3)

This formula has been obtained from the one for QF by replacing Dt,F = Nt,F*(pdt,F)' by Nt,F*pdt,F, and z z by replacing Zt,F = Nships t,F*(pt,F )' by Nships t,F*pt,F .

Unfortunately, we cannot observe the underlying probabilities. Therefore, we take, instead,

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systematic 'lower bounds' for these probabilities.

For this we use a method presented in the paper

by Perepelkin et al. (2010) mentioned above.

Now we show how to get in a systematic way these 'lower bounds'. Let p' be the observed value of the stochast. Thus, p stands for pdt,F rez spectively pt,F , and p' stands for Dt,F /Nt,F respectively Zt,F /Nships t,F. The standard deviation is Let us fix a confidence level a (for example, a = 0.95 is a popular choice) and define for each p the condence interval (p-ca , p+ca) such that a value p' of the stochast is contained in this interval with probability a:

The meaning of this interval is as follows. The values of p that satisfy this condition are the values of p for which the hypothesis that p' lies in the confidence interval of the normal distribution with mean p and standard deviation σ cannot be rejected with confidence level a. To put it more simply, but not entirely precisely: we are 'sure' that the true value of the underlying probability p does not differ more than ca from the observed value p' (here 'sure' means roughly that the probability that this statement is incorrect for given p equals 1-a). We take L(p), the smallest value of p that satisfies this condition as our 'lower bound' of p. The reason for this terminology is that for given p we are confident that L(p) is smaller than p, given the prescribed confidence level a. Because of the fact that p' is normally distributed, it holds that:

where Φ is the cumulative function of the standard normal distribution and Φ(ca /σ) gives the cumulative probability for Z ≤ ca /σ and Z ~ Norm(0,1). It follows that:

Then, we have:

with

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General Now we can define the corrected measure for the performance measure of a flag. We replace the observed value of each stochast (pdt,F)' (and z z (pt,F )') by the 'lower bound' L(pdt,F) (and L(pt,F )) for the mean value of the stochast, as defined above.

This is an inequality in a quadratic polynomial in p. As 1+t2a /N, the coefficient of p2, is positive, the solutions form an open interval with endpoints the roots of the quadratic equation

Then, the abc-formule can be used to get the following 'lower bound' for p:

'final performance measure' (4)

We take this 'lower bound' for QF as our final performance measure for a flag F. Now both shortcomings of the crude performance measure QF' = dF+czF have been overcome: this measure takes into account ship types and it takes into account the variations in the observations.

The other root is an 'upper bound', denoted by U(p). For flags with small fleet, the variation in the p' is larger. Indeed, a good measure for this variation is the difference U(p) - L(p). As the denominator in the expression L(p) (and U(p)) is approximately 2, we see that (U(p) - L(p))2 is approximately (2p'+t2a /N)2-4(1+t2a /N)(p2)'. This can be rewritten as t4a /N2+(2p'ta)/N(ta-p'). As ta â&#x2030;&#x2C6; 2 (usually) and p' â&#x2C6;&#x2C6; (0,1), we see that this is decreasing in N. Thus, for flags with small fleet, the variation is large.

Now we apply this formula to the probabilities of detention and of very serious incidents. After simplication, we get the promised 'lower bounds' of probabilities of detention and very serious incidents for different ship type and different flags:

and

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5. Application of the method to data In the present paper, the focus has been on the formula for measuring performance and how it has been constructed. However a lot more work is reported in the original paper, notably on the application of the proposed method to data.

Here we only mention some points.

We apply our method using inspection (detention) data, incident data and data on the world fleet for the time period 2006 to 2008. We base our analysis on data from Perepelkin et al. (2010) with additional data to extend the time period.

We end up using roughly 183 thousand inspections and 8,646 detentions from various PSC regimes or individual countries (Paris MoU, USCG, Indian Ocean MoU, Vina del Mar Agreement, AMSA) although inspection data are not available for each of these regimes for the entire time period. We further use incident data based on Knapp (2013) from four sources (IHS-Fairplay, Lloyds Maritime Intelligence Unit, International Maritime Organization and the Australian Maritime Safety Authority). For the classification of seriousness, we use the IMO definitions (IMO, 2000) and consider very serious (524 observations) and serious (3,883 observations) incidents. The incident data needed to be manually reclassified in order to ensure compatibility of the four sources. Fleet data was not entirely available for the entire time period by major ship types so we ended up estimating the number of ships for each ship type used (general cargo, dry bulk, container, tanker, passenger vessels and other ship types) and for each flag, based on data from IHS-Fairplay. The following input data are needed to apply the proposed method: • Total number of inspections by ship type and flag • Total number of detentions by ship type and flag • Total number of very serious and total number of serious incidents by ship type and flag • Total number of vessels in service by ship type for each flag • A weight factor for c for very serious incidents and a weight factor d for serious incidents, relative to detention, to be determined by policy makers

than the current method. This has been realized by combining several data sets (and getting these in the first place). This is an advantage.

In the original paper, we report on a number of comparisons of the proposed method with the current method. We try out several weight factors and investigate what happens if we take into consideration serious accidents as well as serious accidents. The upshot of this is that all works well in practice. Now we come to a very sensitive issue: does the ranking of flags changes considerably compared to the current system. This is sometimes the case: flags that are now on the white list become high risk; flags that are on the black list become low risk. No examples can be given here or in the original paper: this issue is politically very sensitive. Nevertheless, we hope that it will be possible to get the proposed fairer measure accepted.

6. Recommendations to policy makers We recommend to policy makers to use the following undesirable events as proxy to determine the effort in enforcing international standards: detentions, serious accidents and very serious incidents. Ship types should be distinguished.

Sympathy should be accorded to make sure that bad luck cannot lead to a ranking of the flag that is too low given its effort. With respect to the use of serious incidents, the method will become more precise once better data is being populated by the IMO via 10 the Global Integrated Ship Information System (GISIS). We also feel that a change in terminology from the current division of Black/Grey/White into High/Medium/Low Risk, where the first two groups are the worst two quartiles, is more appropriate.

Furthermore, we recommend to fix the boundaries between these three groups for three years, based on data from the last three years (possibly use five years' worth of data in the future), and to give flags the opportunity to move upward, especially from High Risk to Medium Risk, in order to reduce occurrence of substandard ships and improve overall safety.

Finally, we recommend that the method is implemented in such a way that no name-and-shame effects can arise for flags from changes in ranks compared to the method currently in use.

The proposed method makes use of more data

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The ballot problem and the waiting time distribution in a non-preemptive queue with priorities Lars van Vianen & Adriana F. Gabor Econometric Institute, Erasmus School of Economics Abstract: In this article we derive a closed form expression of the waiting time distributions in an M/M/c queue with multiple priorities and a common service rate by using a combinatorial approach related to the well known ballot problem. An advantage of the approach is that it relies on purely elementary combinatoric results and does not require inversion of the Laplace Stieltjes Transform.

1. Introduction In many situations where queues arise, prioritizing certain classes of customers leads to an improvement of the system preformance. For example, it is essential to plan the personnel of a hospital such that patients with urgent injuries can be helped fast. In other areas, such as production planning, inventory management and telecommunications, offering faster service to the most valuable clients plays an important role in a succesful business. Due to their relevance in many practical areas, queues with priorities have been intesively studied in the OR literature.

In a priority queueing model, customers are divided in several classes and are offered service according to a prespecified order, dictated by the priority discipline. Most encountered disciplines are the non-preemptive (or head of the line queue) and the preemptive ones. In both, if a service ends, customers with the highest priority in the queue are serviced in order of their arrival (FCFS). In the non-preemptive queue, service of a customer is always completed once it has been started. On the contrary, in the preemptive queue, service is stopped if a customer with

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a higher priority arrives and service is restarted once all the customers with higher priority are served. To evaluate a queuing system, several measures are commonly used, such as the number of customers in the system, the waiting time of customers (the time between the arrival of a customer and the moment when he is offered service) and their response time (the total time a customer spends in system).

The most simple models for queues with priorities assume independent Poisson arrival processes for all customer classes, exponential distributed service times and infinite waiting room. This means that a customer that finds all the servers busy will wait till he can get service.

We will refer to these type of queues as Markovian queues and we will use the Kendall notation M/M/c to refer to their characteristics. Here, the first M indicates the Poisson arrival process, the second M the exponential service time and c the number of servers (or employees who offer service).

In this paper we focus on a non-preemptive M/M/c queuing model with k priority classes of

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Logistics customers who are all served at the same rate.

These type of models are often used for systems where customers get the same service, but have different contracts regarding how fast they will be helped. We show that the distributions of the waiting times can be found by relating the number of customers of different priorities in the system to the well known ballot problem introduced by Bertrand (1887) and to the number of lattice paths between two points in the plane.

Our derivation only involves elementary combinatorial techniques and provides a great simplification of the previously known derivations.

The article is organized as follows. After a review of the related literature in Section 2, we briefly discuss the ballot problem and a few basic results on lattice path enumeration in Section 3.

The distributions of the waiting times are derived in Section 4.

2. Literature Non-preemptive priority queues Non-preemptive priority queues were studied for the first time in Cobham (1953), where the expected waiting times in the M/M/c queue with multiple priorities is derived. The stationary distribution of the number of customers in the system in the two priority non preemptive M/M/1 queue with different service rates is obtained in Miller (1981) by using a Matrix geometric approach. Marks (1973) analyzed a state description for the same queuing model which also includes the priority of the customer which is in service besides the number of customers of each priority.

Research on the waiting times has mainly focused on calculating the Laplace Stieltjes Transform (LST). Based on the LST one can obtain the moments of the waiting times or the waiting time distribution by using numeric inversion techniques (see Abate and Whitt (1992)).

For the single server queue with general service distributions (M/G/1) and two priorities, the LST has been obtained by Kesten and Runnenburg (1957). For the non-preemptive M/M/c queue with multiple priorities and a common service rate μ the Laplace Transform of the waiting time for an arbitrary priority has been derived by Kella and Yechialy (1985) and also by Davis (1966).

Kella and Yechialy observed a close relation with waiting times in an M/G/1 queue with server va-

cations (with one type of customer) and Davis conditioned on the number of customers in the system on arrival with equal or higher priority than the arriving customer.

The waiting time distribution of customers is closely related to the length of busy periods in an M/M/1 queue. A busy period is the time between a customer arrival that finds the server idle and the next moment in time when the server becomes idle again. Dressin and Reich (1956) relate the waiting time in a system with priorities to a convolution of busy periods in a system without priorities and obtain the characteristic function of the waiting time. By inverting this characteristic function, they show that the distribution of the waiting time is given by an infinite sum of Bessel functions.

Combinatorics and the study of Queues Combinatorial techniques have often been used in queuing problems. Tanner (1961) provides a combinatorial proof of the Borel distribution, which gives the distribution of the number of customers participating in a busy period of the M/D/1 queue. Combinatorial techniques were also applied by Takács in many of his works.

Takács (1967) considers the distribution of the supremum of stochastic processes with interchangeable increments, and Takács (1961) derives the joint distribution of the length of a busy period and the number of customers served in the M/G/1 queue. Takács (1962) derives the same joint distribution for the M/G/1 queue where customers arrive in batches of fixed size and also for the G/M/1 queue, making use of a generalization of the ballot problem.

Two more recent examples of combinatoric approaches to queuing problems are Saran and Nain (2013) and Böhm (2010). Saran and Nain derive the transition probability of i arrivals and j departures in an M/M/1 queue during an interval of length t given that there are initially k customers in the system by using results on monotone lattice paths. Böhm (2010), applies recent advancements in lattice paths combinatorics to several queuing models, including systems with bulk arrivals and departures and the preemptive M/M/1 queue with two priorities and a common service rate, where the busy periods is analyzed by using Catalan numbers and Generating Functions.

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3. The Ballot Problem and Monotone Lattice Paths The ballot problem has been introduced by Bertrand (1887). Since then a lot of variations on the initial problem have been analyzed and proven in several distinct ways (e.g. Addario-Berry and Reed (2007), Renault (2007)). The basic problem considered by Bertrand is as follows: There are two politicians who obtained A respectively B votes, with B<A. The ballots are counted one by one. What is the probability ζA,B that the winner has a lead during the entire counting process?

Karlin and Taylor (1981) discuss several solutions to this problem. One of them will be applied in this article to derive waiting time distributions.

This solutions proceeds as follows. Each time a ballot with a vote for the winning (losing) candidate is encountered during the counting process draw a vertical (horizontal) line segment (of unit length) in the plane (start in (0,0)). We obtain a step function which connects (0,0) and (B,A) and will call this a monotone lattice path. An example is given in Figure 1. Clearly each of the (A+B A ) possible monotone lattice paths is encountered with the same probability. However, not all of them correspond to a realization where the winning candidate has always a lead.

Monotone lattice paths where the winner has a lead all the time lie above the line y = 1+x (except for the first vertical line segment). The set of monotone lattice paths from (0,0) to (B,A) which lie above the line l : y = 1+x correspond bijectively to the set of monotone lattice paths between (0,0) and (B,A-1) which lie above the diagonal. Therefore, since the probability of each monotone lattice path (each ordering of the ballots) is the same, we see that ζA,B is equal to the ratio of the number of super diagonal monotone lattice paths between (0,0) and (B,A-1) and the total number of monotone lattice paths between (0,0) and (A,B). Using expressions for these numbers given in Brualdi (2009) Chapter 8, we get:

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(1)

7 6

A

5 4 3 2 1 0 −1 −1

B 0

1

2

3

4

5

6

7

Figure 1: A Monotone Lattics Path with A = 6 and B = 4.

We now define monotone lattice paths and related terminology more formally. A monotone lattice path between two coordinates (a,b) and (c,d), a,b,c,d ∈ ℕ, a ≤ c, b ≤ d is a sequence of disr tinct pairs of integers (ni ,mi ) i=1 , r ∈ ℕ, such that (n1 ,m1) = (a,b), (nr ,mr) = (c,d) which is monotone (that is for a ∈ {n,m} we have ai ≤ aj if i ≤ j). We call each pair in the sequence a node or lattice point. A monotone lattice path is called superdiagonal, if ni ≥ mi for i = 1,..,r.

We will make use of the following result about monotone lattice paths in subsequent proofs.

Lemma 1 (Brualdi (2009), Chapter 8) The number of super diagonal monotone lattice paths between the lattice points (a,b) and (k,k) ≠ (a,b) with a ≤ b, a ≤ k, b ≤ k is given by:

We remark that our terminology is slightly different from the one in Brualdi (2009), where instead of monotone lattice path the term rectangular lattice path is used. Secondly, the results of Brualdi are stated for sub diagonal monotone lattice paths, but the corresponding results for super diagonal elements are easily derived from these.

4. Derivation of Waiting Time Distributions In this section we derive the distribution of the waiting time of a customer of arbitrary priority in the non-preemptive M/M/c queue with K types

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Logistics of customers. We assume that type i priority customers arrive according to a Poisson process with rate λi, i = 1,...,K, where a lower index corresponds to a higher priority. We consider the case where service rates are equal to a common value μ for all types of customers. We will make use of the following additional notation:

To ensure stability, we assume λ < cμ.

Denote the waiting time of priority i customer

by Wi. Consider an arbitrary customer of priority i (which we call the tagged customer). Without loss of generality, we may assume that he arrives at time t = 0. Let Li be the random variable which is equal to zero if c-1 or less servers are busy and equal to n if all servers are busy and n-1 customers of priority i or higher are waiting in queue at the moment when the tagged customer arrives.

Let η0 be the steady state probability ℙ[Li=0] and ηi,n be the steady state probability ℙ[Li=n], which are derived in Davis (1965):

(2)

By conditioning on Li we obtain: (3) Define the process {Δ(s) : s ≥ 0} with state space ℕ, where the state represents the difference between the number of customers in the system that are served before the tagged customer and c-1. We can interpret the state as the number of departures that have to occur before the tagged customer can enter service if no customers of higher priority arrive in between. The state increases upon arrival of a high priority customer and decreases when a service is completed. Note that given Li = n, Δ starts in state n irrespective of the priority composition of the customers that are before him. Note also that the tagged cus-

tomer gets into service when state 0 is hit by Δ.

Define ψ := inf{s : Δ(s) = 0}. We obtain the following result:

(4)

Further, let {Y(s) : s ≥ 0} be a Continuous Time Markov Chain (CTMC) with state space ℤ which has the following properties: given Li = n it starts in state n; it has state independent holding times with common rate γi; lastly the embedded Markov Chain is a simple random walk where each transition is an increase of the state with probability pu := Λi /γi and a decrease of the state with probability pu := cμ /γi. It is clear that Δ can be seen as a restriction of Y to 0 ≤ t ≤ ψ. Following this interpretation the tagged customers gets into service when state 0 is hit for the first time by the process Y. In terms of Y the conditional probability of the event {Wi ≤ t} given L = n becomes:

(5)

Let {τj}∞ j=1 be the sequence of occurrence times of the transitions corresponding to the stochastic process Y. We define for n,k ∈ ℕ the events Bn,k as follows:

Note that the events Bn,k give a partition of the state space. Clearly for k<n the probability of Bn,k is zero, since at least n transitions are required for Y to reach state 0 if it starts at level n. Moreover, since transitions of Y occur according to a Poisson process with rate γi we see that the waiting time of the customer is Erlang distributed with parameters k and γi. Hence, we have:

(6) where Erl(t;k,γi) denotes the cdf of an Erlang random variable with parameters (k,γi) evaluated in t.

Denote the probability mass function of a binomial distribution with parameters n and p evaluated in m by bin(m;n,p). The following lemma ex-

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presses the probabilities ℙ[Bn,k|Y(0)=n] in closed form: Lemma 2

Proof: First, we assert that ℙ[Bn,k|Y(0)=n] is equal to 0 if k-n is not divisible by 2. For n uneven, the state of Y is even if and only if the number of transitions that occurred is uneven. Since 0 is even, this implies that the value of k for which τk = ψ is uneven. Therefore 2 divides n-k. A similar argument holds for the case where n is even. In the remaining of this proof we will only consider the case where n-k is divisible by 2.

We denote a transition of Y by U if the state increases and by D if the state decreases. Let mj be the j-th transition of Y that takes values in the set {U,D}. Furthermore we define a transition sequence (of length r) as a sequence e = {ej}rj=1, with r ∈ ℕ ∪ ∞, ej ∈ {U,D}, j = 1,..,r and define the probability of a transition sequence e by ℙ[e] = ℙ[m1=e1, ..., mr=er]. Let Nu(r) be the number of U transitions and Nd(r) the number of D transitions among the first r transitions of Y. Then, the probability of a transition sequence e of length r is given by:

Similarly, we define the conditional probability ℙ[e|Bn,k] and will say that a transition sequence e is (n,k)-feasible if ℙ[e|Bn,k] > 0. Intuitively, this means that the event sequence represents a realization of the process Y which starts in state n and goes to state 0 in k transitions. Clearly, given Y(0)=n the event Bn,k is determined by the first k transitions of Y (after the first k transitions of Y we know whether Bn,k occurred or not), hence we can compute P[Bn,k|Y(0)=n]as the probability that a transition sequence of length k is (n,k)feasible. The remaining part of this proof consists of this computation.

First, we characterize the set of (n,k)-feasible transition sequences of length k. We assert that the number of D and U events are the same for all such sequences. First, note that Nu(k)+Nd(k) = k. On the other hand, it is also necessary that Nu(k)-Nd(k) = -n since transition k coincides with

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the first time the process Y, which starts at state n, hits state 0. These two relations uniquely determine the number of U and D events. Specifically, we have Nd(k) = 0.5(k+n) and Nu(k) = 0.5(kn). Note that Nd(k) and Nu(k) are integer if an only if k-n is divisible by 2. It follows that each (n,k)-feasible sequence of length k occurs with the same probability (that only depends on n and k). The immediate consequence is that we can compute ℙ[Bn,k|Y(0)=n] by merely counting the number ϱn,k of transition sequences of length k that are (n,k)-feasible, since we have:

The problem of counting the number of distinct (n,k) feasible transition sequences of length k, can be related to counting the number of superdiagonal monotone lattice paths between given lattice points in the plane. First, given a transition sequence {ej}kj=1 we construct a monotone lattice path as follows. We start at the node (0,n1) in the plane, and consider the transitions one by one. For an U transition we draw a vertical line segment, and for a D transition we draw a horizontal line segment. Obviously, we end up in the lattice point with coordinates Nd(k) and n-1+Nu(k). For each (n,k)-feasible transition sequence this ending point is the same. Moreover, there is a bijection between (n,k)-feasible transition sequences and super diagonal monotone lattice paths of length k-1 between the lattice points (0,n-1) and (Nd(k)-1, n-1+Nu(k)). To see this, the following conditions should be satisfied for a transition sequence to be (n,k) feasible: • For r = 1,...,k-1 the number of D transitions among the first r transitions of the process Y exceeds the number of U transitions by at most n-1.

• The first k-1 transitions contain (n+k)/2 - 1 transitions of type D and (k-n)/2 transitions of type U (hence the number of D event lead by n-1).

• Transition k is of type D. Clearly transition sequences that satisfy these conditions and super diagonal monotone lattice paths between the aforementioned lattice points correspond one to one (see also example 1 below).

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Logistics 5 4.5 4 3.5 3 2.5

In van Vianen (2014) we show that this distribution leads to the Laplace Stieltjes Transform derived in Kella and Yechialy (1985). ∎

2 1.5 1 0.5 0

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Figure 2: A Monotone Lattice Path.

Finally, using Lemma 1 on the number of such super-diagonal monotone lattice paths between two lattice points it follows:

5. Concluding Remarks In this article we derived the waiting time distribution in the M/M/c non-preemptive queue with multiple priorities and a common service rate by simple combinatorial arguments using properties of lattice paths. Currently we are exploring similar ideas for finding explicit formula's for the waiting time distributions in other basic queuing models.

References

With this we obtained the desired result, hence the proof is complete. ∎

Example 1 Figure 2 shows the construction of a monotone lattice path from the sequence (DDUDUDD), which corresponds to a (3,7) feasible transition sequence of length 7. Indeed, we see that the first k-1 steps trace out a super-diagonal monotone lattice path between (0,2) and (4,4).

By combining equations (1)-(4) and Lemma 2, we obtain the distribution of Wi: Theorem 1 Consider the M/M/c model with non-preemptive priority and K priority classes.

The waiting time distribution of a priority i customer is given by (where k runs over all integers larger then n which have the same parity as n):

where ηn is given by equation (2) and bn,k and ϱn,k are given by:

[1] Abate, J., Whitt, W., 1992. The Fourier-Series Method for Inverting Transforms of Probability Distributions, Queueing Systems, 10, 5-88.

[2] Addario-Berry L., Reed, B.A., 2008. Ballot Theorems, Old and New, in Bolyai Society Mathematical Studies (17), 9-35.

[3] Bertrand, J., 1887. Solution d'un probleme, Comptes Rendus de l'Academie des Sciences, Paris, 105, 369.

[4] Böhm, W., 2010. Lattice Path Counting and The Theory of Queues, Journal of Statistical Planning and Inference, 140, 2168-2183.

[5] Brualdi, R.A., 2009. Introductory Combinatorics. 5th edition., Prentice-Hall (Pearson).

[6] Cobham, A., 1954. Priority Assignment in Waiting Line Problems, J. Opns. Res. Soc. Am., 2, 70-76.

[7] Davis, R., 1966. Waiting-Time Distribution of a Multi-Server, Priority Queueing System. Opns. Res., 14, 133-136.

[8] Dressin, S.A., Reich. E., 1956. Priority Assignment on a Waiting Line. The Rand Corporation, Santa Monica, Calif., 846.

[9] Karlin, S., Taylor, H.M., 1981. A Second Course in Stochastic Processes, Academic Press.

[10] Kella, O., Yechialy. U., 1985. Waiting Times in the Non-Preemptive M/M/c Queue, Commun. Statist.Stochastic Models, 1(2), 297-262.

[11] Kendall, D., 1951. Some Problems in the Theory of Queues, Journal of the Royal Statistical Society, 13, 152-185.

[12] Kesten, H., Runnenburg, J.T., 1957. Priority in Waiting Line Problems, Proc. Akad. Wet. Amst. A, 60, 312-336.

[13] Marks, B., 1973. State Probabilities of M/M/1 Priority Queues, Operations Research, 21(4), 974-987.

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[14] Miller, D., 1981. Computations of Steady-State Probabilities for M/M/1 Priority Queues, Operations Research, 29(5), 945-958.

[15] Renault, M., 2007. Four Proves of the Ballot Theorem, Mathematics Magazine, 80(5), 345-352.

[16] Saran, J., Nain, K., 2013. Combinatorial Approach to M/M/1 Queues Using Hypergeometric Functions, International Mathematical Forum, 8(10), 463-472.

[17] Takรกcs, L., 1955. Investigation of Waiting Time Problems by Reduction to Markov Processes, Acta Mathematica, Acad. Scient. Hung., 6, 101-128.

[18] Takรกcs, L., 1961. The Probability Law of the Busy Period for Two Types of Queueing Processes, Operations Research, 9(3), 402-407.

[19] Takรกcs, L., 1962. A Generalization of the Ballot Problem and its Application in the Theory of Queues, Journal of the American Statistical Association, 57(298), 327-337.

[20] Takรกcs, L., 1967. On Combinatorial Methods in the Theory of Stochastic Precesses, in Fifth Berkeley Symposium on Mathematical Statistics and Probability (2) , 431-447.

[21] Tanner, J. C. 1961. A derivation of the Borel distribution. Biometrika, 48, 222-224.

[22] van Vianen, L. Waiting Times in Priority Queues and the Ballot Problem, Bachelor Thesis, Erasmus University Rotterdam

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