When analyzing the equilibrium behavior of M/G/1 type Markov chains by transform methods, restrictive hypotheses are often made to avoid technical problems that arise in applying results from complex analysis and linear algebra. It is shown that such restrictive assumptions are unnecessary, and an analysis of these chains using generating functions is given under only the natural hypotheses that first moments (or second moments in the null recurrent case) exist. The key to the analysis is the identification of an important subspace of the space of bounded solutions of the system of homogeneous vector-valued Wiener–Hopf equations associated with the chain. In particular, the linear equations in the boundary probabilities obtained from the transform method are shown to correspond to a spectral basis of the shift operator on this subspace. Necessary and sufficient conditions under which the chain is ergodic, null recurrent or transient are derived in terms of properties of the matrix-valued generating functions determined by transitions of the Markov chain. In the transient case, the Martin exit boundary is identified and shown to be associated with certain eigenvalues and vectors of one of these generating functions. An equilibrium analysis of the class of G/M/1 type Markov chains by similar methods is also presented.
We consider a non-preemptive priority head of the line queueing system with multiple servers and two classes of customers. The arrival process for each class is Poisson, and the service times are exponentially distributed with different means. A Markovian state description consists of the number of customers of each class in service and in the queue. We solve a matrix equation to obtain the generating function of the equilibrium probability distribution by analyzing singularities of the equation coefficients, which are meromorphic matrices of two complex variables. We then obtain the mean waiting times for each class.
We consider a queueing system with multiple servers and two classes of customers operating under a preemptive resume priority rule. The arrival process for each class is Poisson, and the service times are exponentially distributed with different means. In a convenient state space representation of the system, we obtain the matrix equation for the two-dimensional, vector-valued generating function of the equilibrium probability distribution. We give a rigorous proof that, by successively eliminating variables from the matrix equations, a nonsingular, block tridiagonal system of equations is obtained for the set of (m + 1)(m + 2)/2 constants that describe the probabilities of the states of the system when no customers are awaiting service. The mean waiting time of the low priority customers is shown to be given by a simple formula in terms of the (known) waiting time of the high priority customers and the expected number of low priority customers in the queue when no high priority customers are waiting.
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