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International Journal on Communications (IJC) Volume 3, 2014
The Analysis of Performance for Priority Distinction Double-queue and Double-server Communication Network Jialong Xiong*1, Zhijun Yang2, Hongwei Ding3, Qianlin Liu4 School of Information, Yunnan University, Kunming,China
1,3,4
Academy of Educational Sciences, The Education Department of Yunnan Province, Kunming,China
2
*1
xiongjialong2006@126.com; 2yzj@ynjy.cn; 3dhw1964@163.com; 4lql_yn@163.com
Abstract In the field of modern communications network, the quality of a system performance usually determines the application of the system. This paper mainly uses the Markov chain theory to analysis the user blocking rate and losing rate, which are two important indicators of the priority distinction double-queue and double-server communication network(PDDDCN). Through mathematical analysis, we get the corresponding mathematical expression, and make the computer simulations. Simulation results show the correctness of the theoretical analysis, and the relationship between system performance and the arrival rate or the probability that ordinary users fail to relinquish a server. Keywords PDDDCN; Blocking Rate; Losing Rate; System Performance
Introduction In recentlys, there are appeared a lot of new technology of network. It has put forward higher requirements on the performance of the system. When we measure the performance of a system, we always focus on the user blocking rate and losing rate. The cause which leads to these indicators includes the number of servers, the arrival rate of users and the probability that users fail to relinquish a server. This paper assumes there are two servers. We focus on the influence of the arrival rate of primary users and the probability that ordinary users fail to relinquish a server to the quality of the system. By using Markov chain[1,2] theory to analysis this model, we get the relationship between the probability that ordinary users fail to relinquish a server or the arrival rate of primary users and the users blocking rate or users losing rate. Finally, in this paper we use the MATLAB to simulate the model. The simulation results prove the correctness of the theoretical analysis.
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The description of the PDDDCN model In this paper, we assume that there are two servers, and the two queues having different priorities [3,4]. The users who have high priority are defined as primary users (PUs). The users who have low priority are defined as Ordinary users (OUs). This model assumes that the arrival process of PUs and OUs are Poisson process and the service process to PUs and OUs are exponential process. The arrival rate of PUs is λ p and OUs is λc . The service rate of PUs is µ p and OUs is µc . When a PU arrives and the server is idle, the PU directly uses the server. However, when a PU arrives and the server isn't idle, The PU may or may not get service. Assuming that servers are used by PUs, the new arrival PU cannot be service. But if one of the two servers is used by the OU, the new arrival PU maybe can get service. Here we assume that the PU gets service at rate λ j ( j means that j servers are used by the OU), the probability of OU failing to recognize the PU arrival (or ordinary users fail to relinquish a server) is q ( 0 ≤ q ≤ 1 ), so λ j= (1 − q j ) λ p . If p j = 1 − q j , so λ j = p j λ p , and we can define that p j is the prior probability of the PU. FIG.1 is the model of the PDDDCN. (1) When q = 1 , it demonstrates that The OU completely fail to recognize the PU arrival. If the server is being used by OU, the PU doesn’t have the right to use the server. (2) When 0 < q < 1 , it demonstrates that the PU has partial priority. If two servers aren't idle and one server is or two servers are used by the OU, the PU has priority with probability p j to use the server. (3) When q = 0 , it demonstrates that the PU has super
International Journal on Communications (IJC) Volume 3, 2014
priority. If two servers aren't idle and one server is or two servers are used by the OU, The OU immediately releases the server, and the server provides service to PU. The OU will drop from the queue. Service rate µ p
PUs λ p
Server 2
When two servers aren't idle and one server is or two servers are used by the OU, the new arrival PU may or may not get service from the server. If the probability of OU failing to recognize the PU arrival is q , the probability of new arrival PU get service from the server is p j = 1 − q j , there 0 ≤ q ≤ 1
FIG.1 The PDDDCN model
The state transition model of the PDDCN From the description of Performance for PDDDCN, and comparing with the M/M/1 queue [5], we can obtain a conclusion that the state transition process of the model is consistent with the Markov process. Assume that the system is staying in a stable state. State ( i, j ) means that the state of the server. Let
pi , j
denote
the
steady-state
probability
of
state ( i, j ) , i means that i servers are used by the PU, j means that j servers are used by the OU, where i, j = 0,1, 2 , and 0 ≤ i + j ≤ 2 . FIG.2 is the state transition model of the PDDDCN. Horizontal flows to right and left represent arrivals of PUs and departures of PUs. Vertical flows to up and down represent arrivals of OUs and departure of OUs. When the system is staying ( i, 2 − i ) , it represents a full system. If q ≠ 0 and i = 0,1 , we should consider what happens when a PU arrives and the system is in state ( i, 2 − i ) . The state ( i, 2 − i ) will move to state ( i + 1,1 − i ) . From the description of the model, we can get the state transition rate is λ j , λ j= (1 − q j ) λ p . 0, 2
λ2 2 µc
λc
λp
0,1
1,1
λ1
µp
µc
λc
µc
λc λp
0, 0
λp 2, 0
1, 0
µp
equation [6,7] to calculate the transition probability, we can obtain the flow balance equations. The analysis proceeds by solving " flow in=flow out " for each of the states.
( 0, 0 ) , ( 0,1) , ( 0, 2 ) , (1, 0 ) , (1,1) ,
For state
( 2, 0 ) , the flow balance equations are (1)-(6).
Server 1
Service rate µc
OUs λc
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2µ p
FIG.2 The state transition model of the PDDDCN
The analysis of performance for PDDDCN According to the method of Chapman-Kolmogorov
p0,0 (λ p + λc = ) p1,0 µ p + p0,1 µc
(1)
p0,1 ( µc + λ p + λc= ) p1,1 µ p + 2 p0,2 µc + p0,0 λc
(2)
p0,2 (2 µc + λ2 ) = p0,1λc
(3)
p1,0 ( µ p + λ p + = λc ) 2 p2,0 µ p + p1,1 µc + p0,0 λ p
(4)
p1,1 ( µ p + µc + λ1= ) p0.2 λ2 + p0,1λ p + p1,0 λc
(5)
= µ p p1,1λ1 + p1,0 λ p 2 p2,0
(6)
Where λ1 = (1-q ) λ p , λ2 = (1-q 2 ) λ p . The steady-state probabilities sum to unity so that we also have 2 2− j
∑∑ p
i, j
=j 0=i 0
(7)
=1
Making the equations (1)-(5) and (7) are written in vector form, the can be written as λ p + λc −λc 0 −λ p 0 1
− µc µc + λ p + λc −λc
0 −2 µc
0
2 µc + λ2 0
−λ p 1
−λ2 1
−µ p 0
0 −µ p 0
0
µ p + λ p + λc −λc
− µc
µ p + µc + λ1
1
1
0 p0,0 0 0 p0,1 0 0 p0,2 0 = −2µ p p1,0 0 0 p 0 1,1 1 p2,0 1
(8)
Let us denote λ p + λc −λc 0 M = −λ p 0 1
− µc µc + λ p + λc −λc
P = ( p0,0
0
0 −2 µc 2 µc + λ2 0
−µ p 0 0 µ p + λ p + λc
0 −µ p 0 − µc
−λ p 1
−λ2 1
−λc 1
µ p + µc + λ1
p0,1
p0,2
p1,0
1
0 0 0 −2µ p 0 1
p1,1
p2,0 )
T
B = ( 0 0 0 0 0 1)
T
The (8) can be written as (9) MP = B The steady-state probabilities can now be computed using (9) as (10) P = M −1 B After calculating the steady-state probabilities of each state of the system, we can compute the users blocking rate and users losing rate. The user blocking rate is defined as probability, that when a user arrive, but the server is being used by other users, resulting in the new user can't immediately use server. In this system, it can be divided into PU blocking rate and OU blocking rate. The user losing rate is defined as a ratio of losing rate and receiving rate. In this system, it also can be divided into PU losing rate and OU losing rate. From the FIG.2, we can get that the PU losing rate is 0 all the time, but the OU losing rate is not always 0.
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International Journal on Communications (IJC) Volume 3, 2014
Only p = 0 , the OU losing rate is 0.
q=0 0.35
From the above definitions, we can get: PU blocking rate is 2
+∑ p
P = p
(1 − p ) = p
The theoretical value of PU blocking The simulation value of PU blocking
0.3
The simulation value of OU blocking
2
+ ∑ p2 − j , j q
The theoretical value of OU blocking
(11)
j
Probability
2,0 2− j , j 2,0 b1 j =j 1 =j 1
OU blocking rate is 2
Pb 2 = ∑ pi ,2 − i
The theoretical value of OU losing The simulation value of OU losing
0.25
(12)
0.2
0.15
0.1
i =0
OU losing rate is 2
∑p
0.05
2
λ
∑p
2− j , j j =j 1 =j 1 L c b2 c
P=
λ (1 − p )
=
2− j , j
0
λj
(13)
2
λ (1 − ∑ pi ,2 −i ) i =0
The simulation of the PDDDCN model From above analysis, this paper does a MATLAB simulation for the model. First, we assume that the model is staying the ideal state, and the q will not change. We discuss that the relationship between model performance and PUs arrival rate. Let λC = 1 ,
µc = 3 , µ p = 2 . FIG.3 to FIG.5 give the blocking and losing probabilities of the PUs and OUs changing with the arrival rate of PUs, when q is different. q=1 0.35 The theoretical value of PU blocking The simulation value of PU blocking
0.3
The theoretical value of OU blocking The simulation value of OU blocking
Probability
0.2
0.15
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
λp
FIG.5 When q = 0 , the blocking and losing probabilities of the PUs and OUs change with the arrival rate of PUs.
From FIG.3 to FIG.5, we can obtain that the blocking and losing probabilities of the PUs and OUs are increasing with the increase of the PUs arrival rate. When q = 0 , the PU has super priority, so the PUs blocking rate is less than the OUs blocking rate. When q = 0.35 , the PU has partial priority, the PU's arrival would force the OU to relinquish the use of server with probability 1 − q j , so the PUs blocking rate is also less than the OUs blocking rate. When q = 1 , the OU completely fail to recognize the PU arrival, it demonstrates that the PU and OU have equal priority, so the blocking rate of the PUs and OUs are equal and the losing rate of PUs is 0.
µc = 3 , µ p = 2 . FIG.6 gives the blocking and losing
0.1
probabilities of the PUs and OUs changing with the probability q .
0.05
0
0.2
Second, we assume that the model is staying the ideal state, and the PUs arrival rate will not change. We discuss that the relationship between model performance and probability q . Let λC = 1 , λ p = 1 ,
The theoretical value of OU losing The simulation value of OU losing
0.25
0
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
λp
0.18
FIG.3 When q = 1 , the blocking and losing probabilities of the PUs and OUs change with the arrival rate of PUs.
0.16 0.14
q=0.35 0.12
0.35 Probability
The theoretical value of PU blocking The simulation value of PU blocking
0.3
The theoretical value of OU blocking The simulation value of OU blocking The theoretical value of OU losing The simulation value of OU losing
0.25
0.1 0.08 0.06
Probability
The theoretical v alue of PU blocking
0.2
0.04
The theoretical v alue of OU blocking The simulation v alue of OU blocking
0.02
The theoretical v alue of OU losing
0.15
The simulation v alue of OU losing
0
0.1
0
0
0.1
0.2
0.3
0.4
0.5 q
0.6
0.7
0.8
0.9
1
FIG.6 When λ= λ= 1 ， µC = 3 ， µ p = 2 , the blocking and losing c p
0.05
probabilities of the PUs and OUs change with the probability q . 0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
λp
FIG.4 When q = 0.35 , the blocking and losing probabilities of the PUs and OUs change with the arrival rate of PUs.
60
The simulation v alue of PU blocking
From the FIG.6, we can obtain that the PUs blocking rate is increasing with the increasing of probability q . And the OUs blocking rate and OUs
International Journal on Communications (IJC) Volume 3, 2014
losing rate are decreasing with the increasing of probability q . The simulation results are consistent with the theoretical results.
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[6] R. N. Miroshin. On some solutions of ChapmanKolmogorov
equation
for
discrete-state
Markov
processes with continuous time[J].Vestnik St. Petersburg University: Mathematics, 2010,43(2):63-67.
Summary In the field of modern communications network, the quality of a system’s performance usually determines the application prospect of the system, and then it’s very important to analysis the performance of a communications system model. This paper mainly uses the Markov chain theory to study the PDDDCN model from the user’s blocking rate and losing rate. And we use the MATLAB to simulate the model’s theoretical analysis. The simulation results show that the theoretical analysis is correct, at the same time, it’s proves the PUs arrival rate and the probability of OU failing to recognize the PU arrival have reflects on the model’s performance. In this paper, this model also demonstrates the advantages of the priority distinction service, and with the development of technology, only two server cannot satisfy the user’s requirement. In the future, our research focus will be the multi-server model. ACKNOWLEDGMENT
Foundation item: National Natural Science Foundation funded project (61072079), Yunnan Provincial Natural Science Foundation Project (2010CD023), Yunnan University financial support project (No.XT412004).
[7] Eldad P. IEEE 802.11n development: History, process and technology[J]. IEEE Commun. Mag.,2008, 46(7):48-55. Jialong Xiong (1988-), male, Tengchong Yunnan, a master student of Yunnan University, 2010 graduated from School of Information Science and Engineering, Yunnan University, The main research directions are computer communication network and Polling systems theory. Zhijun Yang (1968-), male, Baoshan Yunnan, Doctoral degree, Vice president of Education Science Institute of Yunnan Province, The main research directions are communication and information system and communication network. Hongwei Ding (1964-), male, Yudu Jiangxi, Doctoral degree, associate professorthe, The main research directions are random multiple access communication system. Qianlin Liu (1968-), male, Qujing Yunnan, Doctoral degree, Senior Engineer, The main research directions are Polling systems theory and computer communication network.
REFERENCES
[1] Demirkol I, Ersoy C, Alagoz F. MAC protocols for wireless sensor networks:a survey[J]. IEEE Commun. 2006, 44(4):115-121 [2]
Craven,Bruce
D.Markov
Chains
and
Invariant
Probabilities[J]. SIAM Review, 2005, 47(4):819-821. [3] Liu Qianlin, Zhao Dongfeng, Zhao Yifan. An efficient priority service model with two-level-polling scheme[J]. High Technology Letters, 2011, 17(3): 245-251. [4] Zhao Dongfeng, Ding Hongwei, Zhao Yifan, Wang Mingui. An Analytical Model of a Discrete-Time Polling MAC Protocol for Wireless LANs Using M-Gated Services[J]. Acta. Electronica Sinica, 2010, 38(7): 14951499. (in Chinese) [5] Applied Mathematics. Server Workload in an M/M/1 Queue with Bulk Arrivals and Special Delays[J]. Applied Mathematics, 2012, 3(12):2174-2177.
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