Queueing Analysis of a Jockeying Model

In this paper, we solve a type of shortest queue problem, which is related to multibeam satellite systems. We assume that the packet interarrival times are independently distributed according to an arbitrary distribution function, that the service times are Markovian with possibly di erent service rates, that each of the servers has its own bu er for packet waiting, and that jockeying among bu ers is permitted. Packets always join the shortest bu er(s). Jockeying takes place as soon as the di erence between the longest and shortest bu ers exceeds a pre-set number (not necessary 1). In this case, the last packet in a longest bu er jockeys instantaneously to the shortest bu er(s). We prove that the equilibrium distribution of packets in the system is modi ed vector-geometric. Expressions of main performance measures, including the average number of packets in the system, the average packet waiting time in the system and the average number of jockeying, are given. Based on the above solutions, numerical results are computed. By comparing the results for jockeying and non-jockeying models, we show that a signi cant improvement of the system performance is achieved for the jockeying model. The performance study of a great number of satellite systems basically depends on the analysis of the related queueing systems. The major interesting measures of such analysis includes the system throughput, the average packet delay on the satellite, and the bu er over ow probability for the case of nite bu er size. Multibeam satellite systems have been studied extensively (for example, see Chlamtac and Ganz 1986, and Chang 1983), and it has been shown that they provide a greater system exibility and a better performance. In such a system, all earth stations are organized into disjoint zones; packets generated from earth zones arrive at the satellite by using di erent possible access techniques, one or several bu ers are provided at the satellite for the waiting packets to be processed or transmitted; and nally, the packets are sent to their destinations by the multi-down-link beams. When there is more than one bu er on board, introducing jockeying of the waiting packets among the bu ers seems to be a promising way to improve the performance of the systems. For example, if we allow a packet waiting in the bu er with many waiting packets to move to some other bu er with fewer waiting packets in it, then the average packet waiting time is obviously reduced. But the analysis of such jockeying systems is more di cult because we cannot deal with them by analysing only one speci ed input-output pair. Instead, we must handle the system as a whole. According to di erent assumptions made on the system, a variety of di erent jockeying models could arise. In this paper, we consider a very general type of jockeying model, in which the following assumptions are made. The arriving packets follow a general process; that is, the time between any two successive arriving packets is described by an arbitrary non-negative random variable, the interarrival time. All the interarrival times are identically and independently distributed. There are several bu ers in the system, each of them with an in nite capacity. An arrival packet always joins the shortest waiting line if it cannot be processed immediately. For any waiting line, the waiting packets are processed according to rst come rst served (FIFO) discipline. The processing time for any packet is a Markovian, that is, an exponential random variable. When the di erence of the waiting packet numbers between the longest waiting line and the shortest one exceeds a certain threshold value, the last waiting packet is allowed to move (jockey) to the shortest waiting line. Since Haight (1958) proposed and solved the shorter queue model (the shortest queue model with only two servers), the jockeying problem has been studied extensively, in particular 1 by Disney and Mitchell (1971), Elsayed and Bastani (1985), Kao and Lin (1990), Zhao and Grassmann (1990), Zhao (1990), and Adan, Wessels and Zijm (1991). Except Zhao and Grassmann, all authors considered only models with Markovian inputs. Among them, Kao and Lin solved the problem of jockeying as soon as the di erence between queues exceeds one. They expressed their solution in terms of the eigenvalue of the rate matrix. Using the results of Kao and Lin, Zhao and Grassmann (1990) developed an explicit solution to the problem. They showed how to obtain certain initial probabilities for the system, and expressed the joint distributions of the queue lengths in the vector-geometric form. Recently, Nelson and Philips (1989) studied the response time for shortest queue routing by using approximations. As stated in their paper, shortest queue routing is a natural way to balance the load of a system across several processors and has been used as a load balancing mechanism as well as a scheduling mechanism in an e ort to minimize job response time. In this paper, we consider the general input model with a very exible jockeying rule, in which the last packet in the longest queue jockeys to the shortest queue with an arbitrary probability distribution as soon as the di erence of the waiting packet numbers between the longest queue and the shortest queue exceeds r, r 1. Some special cases of this model have been considered by Zhao in his Ph.D thesis (Zhao 1990). After giving the de nition of the model in the next section, we rst consider the imbedded Markov chain of the model. We then obtain an explicit solution of the model, which also, as expected, has a vector-geometric form. Other interesting system measures are given, based on the probability distribution of the packet's number in the system. Numerical results are presented and analyzed. It turns out that signi cant improvements of the system performance can be achieved by allowing jockeying among the queues. 1 The Model and the Imbedded Markov Chain In this section, we give the de nition of the r di erence jockeying problem with a general input, and point out some important properties of the transition probabilities. In order to give a formal de nition of the shortest queue model with r di erence jockeying, we make the following assumptions: a) The packets arrive singly with interarrival times identically and independently distributed 2 according to an arbitrary distribution function A(t) with A(t) = 0 if t < 0, and they are not allowed to renege or balk. b) There are c (c 2) servers (for example, transponders or down-link beams), numbered 1; 2; : : : ; c, in the system and each of them has its own bu er. In each bu er, service is rendered according to FIFO ( rst come rst served) discipline. The c servers have independent exponential service times. The service times are independent of arrivals. c) An arriving packet joins one of the shortest bu ers with a pre-determined probability distribution. d) Jockeying among the bu ers is permitted. The last packet in the longest bu er(s) instantaneously jockeys to the shortest bu er(s) with a pre-determined probability distribution as soon as the di erence of the packet numbers between the shortest bu er(s) and the longest bu er(s) exceeds r, r 1. A queueing system satisfying a) { c) is called the shortest queue model and denoted by GI=(M=1)c. A shortest queue model satisfying d) is called the shortest queue model with r di erence jockeying. When r = 1, we simply call it the shortest queue model with jockeying. When both the probability distributions mentioned in c) and d) are uniform, we call the shortest queue model symmetric; otherwise non-symmetric. Let Xk(t) represent the number of waiting packets in bu er k, k = 1; 2; : : : ; c, at time t, t 0, including the packet in service, then f ~ X(t) = (X1(t);X2(t) , : : : ;Xc(t)) ; t 0 g is a stochastic process. The state space of this process can be described as S = f~i = (i1; : : : ; ic) j ij non-negative integer for j = 1; 2; : : : ; c and jik ilj r for k; l = 1; 2; : : : ; c g : The main purpose of this paper is to determine the limiting probabilities ~i = lim t!1Pf ~ X(t) =~i g ; ~i 2 S ; when they exist. In general, the above process is neither Markovian nor semi-Markovian. In order to analyze this model, we introduce the imbedded Markov chain for the system as follows. We obtain 3 an explicit formula for the probability distribution of bu er lengths (including the packets in service) for the imbedded Markov chain rst. The probability distribution of the bu er lengths at a random time can then be found from the connection between a semi-Markov process and its imbedded Markov chain. If tl is the time just prior to the arrival of the lth packet, then f ~ Xl = (X1(tl);X2(tl), : : : ;Xc(tl)) ; l = 1; 2; : : : g is Markovian. Let 1= be the mean time between two successive arrivals, and let be the sum of all the service rates; that is, = Pck=1 k. The imbedded Markov chain f ~ Xl ; l = 1; 2; : : : g formed in this way is ergodic if, and only if, the tra c intensity = is less than one. In the paper, we always assume that this is the case. In the stable condition, the limiting or equilibrium probabilities p~i = lim l!1Pf ~ Xl =~ig ; ~i 2 S ; exist and they are the same as the stationary probabilities of the imbedded Markov chain. For any two states ~i, ~j 2 S, the transition probability p~i~j can be found by conditioning on the interarrival time Ul; that is, p~i~j = Z 1 0 Pf ~ Xl+1 = ~j j Ul = t ; ~ Xl =~i g dA(t) : An explicit determination of p~i~j can be obtained by using conditional probability arguments. Since only elementary algebraic manipulations are involved for the derivation and the nal explicit expression of p~i~j is cumbersome, we will not produce it here. The readers, who are interested in details of deriving the explicit expression of p~i~j, may refer to Zhao (1991), in which the same technique was used for obtaining the expression of p~i~j for the case of r = 1. Instead, we give the proofs, by using the same conditional probability argument, of the following properties of the transition probabilities. These properties are essential for proving our main results in this paper. For a state ~i, de ne #~i to be the number of packets in the system; that is, if ~i = (i1; i2; : : : ; ic) then #~i = Pck=1 ik, and de ne two functions l( ) and s( ) of states to be the number of packets in the longest bu er and the shortest bu er respectively; that is, l(~i) = max(i1; i2; : : : ; ic) and s(~i) = min(i1; i2; : : : ; ic). De ne ~1 = (1; 1; : : : ; 1). 4 Proposition 1 Let ~i, ~j 2 S be two states in the state space. If #~j > #~i+ 1, then p~i~j = 0. If #~j = #~i+ 1 and s(~j) > 0, then p~i~j = p~i+~1 ~j+~1, and X~j: #~j=#~i+1 p~i~j = 0 ; (1) where 0 = Z 1 0 e tdA(t) : (2) If #~j < #~i+ 1 with l(~j) > r, de ne k to be (#~i+ 1) #~j. Then p~i~j = p~i+~1 ~j+~1, and X~j: #~j=#~i+1 k p~i~j = k ; (3) where k = Z 1 0 ( t )k k! e t dA(t) : (4) First, it is obvious that if #~j > #~i+ 1, then p~i~j = 0. In the following we de ne ~ X 0 l to be the state immediately after the arrival of the lth packet. If #~j = #~i+ 1 and s(~j) > 0, then p~i~j = Z 1 0 Pf ~ X 0 l = ~j j ~ Xl =~i gPf ~ Xl+1 = ~j j Ul = t ; ~ X 0 l = ~j g dA(t) = Z 1 0 Pf ~ X 0 l = ~j j ~ Xl =~i gPf no packets served j Ul = t ; all servers busy at tl gdA(t) = Z 1 0 Pf ~ X 0 l = ~j + ~1 j ~ Xl =~i+ ~1 ge t dA(t) = p~i+~1 ;~j+~1 ; (5) where ~1 = (1; 1; : : : ; 1). Notice that whether p~i~j > 0 or p~i~j = 0 depends on whether Pf ~ X 0 l = ~j j ~ Xl =~i g is greater 0 or equal to 0. Also notice that the condition s(~j) > 0 is weaker than s(~i) > 0. It follows from (5) and P~j Pf ~ X 0 l = ~j j ~ Xl =~i g = 1 that X~j: #~j=#~i+1 p~i~j = Z 1 0 e t dA(t) = 0 : (6) 5 If #~j < #~i+ 1 with l(~j) > r, de ne k = (#~i+ 1) #~j. If ! ~i1 !~i2 ! ~ik is de ned as the event that the system is in state ~i1 after the rst packet served, in state ~i2 after the second packet served, ... , and in state ~ik after the kth packet served, then, by using l(~j) > r, p~i~j becomes p~i~j = Z 1 0 X ~i1;:::;~ik:E Pf !~i1 !~ik = ~ Xl+1 = ~j j Ul = t ; ~ Xl =~i g dA(t) = Z 1 0 X ~i1;:::;~ik:E Pf !~i1 + ~1 !~ik + ~1 = ~ Xl+1 = ~j + ~1 j Ul = t ; ~ Xl =~i+ ~1 g dA(t) = p~i+~1 ;~j+~1 ; (7) where E denotes the event f#~i1 = #~i; #~i2 = #~i 1, . . . , #~ik = #~i k = #~jg, and ~1 = (1; 1; : : : ; 1). It follows from (7) that X~j: #~j=(#~i+1) k p~i~j = Z 1 0 X ~i1 ;:::;~ik:E Pf !~i1 !~i2 !~ik j Ul = t ; ~ Xl =~i gdA(t) = Z 1 0 Pf k customers served j Ul = t ; all servers busy at tl gdA(t) = Z 1 0 ( t )k k! e t dA(t) = k : (8) Notice that l(~j) > 0 is only a su cient condition for the above property. One may prove (8) under a weaker condition. 2 Solution of the Imbedded Markov Chain This section shows that the equilibrium probabilities of packet lengths for the imbedded Markov chain of the model under consideration obey a distribution which we call vectorgeometric. By this, we mean that there is a constant, say !, such that every state, except for boundary states, is related to exactly one other state in the sense that the ratio between their probabilities is !. To do the proof, we introduce the concept of blocks of states, and the concept of groups of states. The proof exploits the special structure of the transition matrix. 6 Speci cally, some states cannot be reached by an arrival. Moreover, after an arrival, the states in block k + 1 can be reached from block k only through state (k; k; : : : ; k). The proof also exploits the connection between the imbedded Markov chain of the GI=(M=1)c model with r di erence jockeying and the imbedded Markov chain of the GI=M=1 model. In order to state our main result, we need to partition the state space S into blocks according to the maximal number of packets in the bu ers. Let B>>>=>>>>; I The equations given in I(a) and I(b) are called boundary equations, and those in I(c) queue equations. Main result: The solution of the stationary probabilities of the imbedded Markov chain has a modi ed vector-geometric form. Speci cally, ~ pm+1 = c~ pm ; m = r + 1; r + 2; : : : ; (13) and (~p>>>>>>>>=>>>>>>>>; II Our aim is to prove that system II has a non-trivial solution for ~ p t ; ~ X(tn) =~i g [ 1 A(t) ]dt = X~i p~i Z 1 0 Pf ~ X(tn+1) = ~j j Un = t ; ~ X(tn) =~i g [ 1 A(t) ]dt : For a proof of the rst equality, use fact (iv) on page 351 of Gross and Harris and the fact that the limiting probability distribution of the imbedded semi-Markov chain is the same as that of the imbedded Markov chain since the mean time spent in every state during a visit is the same equal to 1= . The proof of the second equality is based on the memorylessness of the service times and the independence of the service times and the interarrival times. Under these conditions, the transition probability from ~i at the previous imbedded epoch to ~j is independent of t the length of the time. Partition the equilibrium probability vector ~ according to blocks de ned on the state space: ~ = ( ~ Y. Zhao acknowledges that this research was supported by a grant from the CanadianInstitute for Telecommunications Research under the NCE program of the Government ofCanada when he was with Queen's University andW.K. Grassmann acknowledges the researchsupport through a research grant of the NSERC. The both authors gratefully thank ProfessorL.L. Campbell for reading the manuscript and suggesting a comparison between jockeyingand non-jockeying models, and referees for their valuable suggestions and comments whichimproved the presentation of the paper.22 ReferencesAdan, I.J.B.F., J. Wessels and W.H.M. Zijm. 1991. Analysis of the asymmetric shortestqueue problem with threshold jockeying. Stochastic Models 7(41), 615{28.Chang, J.F. 1983. A multibeam packet satellite using random access techniques. IEEETrans. Commun. COM-31, 1143{1154.Chaudhry, M.L., M. Agarwal and J.G.C. Templeton. 1992. 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Performance Evaluation Review 17(1), 181{189.Neuts, M.F. 1981. Matrix-Geometric Solutions in Stochastic Models. Johns Hopkins Univer-sity Press, Baltimore.Zhao, Y. and W.K. Grassmann. 1990. A solution of the shortest queue model with jockeying{23 in terms of tra c intensity . Naval Research Logistics 37, 773{787.Zhao, Y. 1990. Shortest Queue Models. Ph.D. Thesis, University of Saskatchewan.24 0123456Number of packets in the system0.010.070.130.190.250.31Probabilitiesnon-jockeyingjockeying with r=1 Figure 1: The equilibrium probabilities of the number of packets in the system for the shortestqueue model M=(M=1)2 with the tra c intensity = 0:5 and equal service rates.25 0.40.480.560.640.720.8Traffic intensity1.62.22.83.444.6 AveragepacketwaitingtimeinthesystemFrom top to bottom:non-jockeying, r=2 and r=1. Figure 2: The average packet waiting time in the system for the shortest queue modelM=(M=1)2 with the ratio of the service rates 9 to 1.26 0.40.480.560.640.720.8Traffic intensity1.31.72.12.52.93.3AveragepacketwaitingtimeinthesystemFrom top to bottom:non-jockeying, r=2 and r=1. Figure 3: The average packet waiting time in the system for the shortest queue modelM=(M=1)2 with the ratio of the service rates 3 to 1.27 0.40.50.60.70.80.9Traffic intensity1.122.93.84.7Averagepacketwaitingtimeinthesystemnon-jockeyingjockeying with r=1 Figure 4: The average packet waiting time in the system for the shortest queue modelM=(M=1)2 with the equal service rates.28