Markovian Queueing Systems
Queueing systems whose underlying stochastic process is a continuous-time Markov chain (CTMCs) are the simplest and most often used class of queueing systems. The analysis of these systems is based on the essential results available for the analysis of CT
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Markovian Queueing Systems
Queueing systems whose underlying stochastic process is a continuous-time Markov chain (CTMCs) are the simplest and most often used class of queueing systems. The analysis of these systems is based on the essential results available for the analysis of CTMCs. As a consequence, several interesting properties of these queueing systems can be described by simple closed-form analytical expressions both in transient (as a function of time and initial state) and in steady state. The most often studied property of basic queueing systems is the queue length process, N.t/; t 0, which represents the number of customers in the system at time t. If customers belong to K different customer classes, then the vector-valued function N.t/ D .N1 .t/; : : : ; NK .t//, t 0, describes the queue length process. In this case the i th component of N.t/, Ni .t/, is the number of class i customers in the system. The queue length process of basic queueing systems with a single class of customers is the birth-death process, which is a special CTMC. We will utilize the previously introduced results of CTMCs and birth-death processes for the analysis of queueing systems.
7.1 M=M=1 Queue The most basic queueing system is the M=M=1 queue, which is composed of a single server and an infinite buffer. Figure 7.1 identical customers arrive according to a time-homogeneous Poisson process at rate , and the service time of a customer is exponentially distributed at rate . The customers are served in the order of their arrival (FCFS). The server is always busy as there is at least one customer in the system. This last property is referred to as a work-conserving property, which we commonly assume in the sequel unless otherwise stated. Let N.t/ be the number of customers in a system (either being served or waiting in the buffer) at time t. Due to the memoryless property of the arrival and the service process, N.t/ is a CTMC (Fig. 7.2). Its (nonvanishing) state-transition probabilities L. Lakatos et al., Introduction to Queueing Systems with Telecommunication Applications, DOI 10.1007/978-1-4614-5317-8 7, © Springer Science+Business Media, LLC 2013
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7 Markovian Queueing Systems
Fig. 7.1 M=M=1.=1/ queueing system
Customer arrivals: λ with rate Poisson process
λ 0
μ
λ 1
λ
λ 2
μ
Service time distribution: Exponetial with parameterμ
3
μ
μ
λ 4
μ
...
Fig. 7.2 Markov chain of number of customers in M=M=1 queue
are pi;i C1 ./ D C o ./ ;
i D 0; 1; : : : ;
pi;i 1 ./ D C o ./ ;
i D 1; 2; : : : ;
pi i ./ D 1 . C / C o ./ ;
i D 1; 2; : : : ;
p0;0 ./ D 1 C o ./ ; where pi;j .t/ D P .N.t/ D j j N.0/ D i /. That is, N.t/ is an infinite state birthdeath process with i D .i D 0; 1; : : :/ and i D .i D 1; 2; : : :/. The Markov chain is irreducible, and from its stationary equations we have i D
i ;
i D 0; 1; : : : :
According to Eq. (3.19) this Markov chain is stable if < 1; that is, < : The intuitive explanation of this relation is straightforward. It means t
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