Sidney L. Hantler

Spectral analysis of M/G/1 and G/M/1 type Markov chains

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.

Analysis of a non-preemptive priority multiserver queue

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.

Non-Skip-Free M/G/ 1 and G/M/ 1 Type Markov Chains

For Markov chains of M/G/ 1 type that are not skip-free to the left, the corresponding G matrix is shown to have special structure and be determined by its first block row. An algorithm that takes advantage of this structure is developed for computing G. For non-skip-free M/G/ 1 type Markov chains, the algorithm significantly reduces the computational complexity of calculating the G matrix, when compared with reblocking to a system that is skip-free to the left and then applying usual iteration schemes to find G. A similar algorithm to calculate the R matrix for G/M/ 1 type Markov chains that are not skip-free to the right is also described.

Solutions of the basic matrix equation for M/G/l AND G/M/1 type markov chains

Let be a sequence of nonnegative matrices such that is a substochastic matrix. The unique minimal nonnegative solution of the matrix equation has been shown by M. F. Neuts to play a key role in the analysis of M/G/l type Markov chains. In this paper, all of the power-bounded, matrix solutions of this equation are classified. Among these solutions, the subsets of nonnegative, substochastic and stochastic solutions are identified. In particular, the exact conditions under which the equation has infinitely many power-bounded solutions (infinitely many stochastic solutions) are given. Similar results are obtained for the solutions of the matrix equation which appears in the analysis of G/M/l type Markov chains

Use of Characteristic Roots for Solving Infinite State Markov Chains

In this chapter, our interest is in determining the stationary distribution of an irreducible positive recurrent Markov chain with an infinite state space. In particular, we consider the solution of such chains using roots or zeros. A root of an equation f (z) = 0 is a zero of the function f (z),and so for notational convenience we use the terms root and zero interchangeably. A natural class of chains that can be solved using roots are those with a transition matrix that has an almost Toeplitz structure. Specifically, the classes of M/G/1 type chains and G/M/1 type chains lend themselves to solution methods that utilize roots. In the M/G/1 case, it is natural to transform the stationary equations and solve for the stationary distribution using generating functions. However, in the G/M/1 case the stationary probability vector itself is given directly in terms of roots or zeros. Although our focus in this chapter is on the discrete-time case, we will show how the continuous-time case can be handled by the same techniques. The M/G/1 and G/M/1 classes can be solved using the matrix analytic method [Neuts, 1981, Neuts, 1989], and we will also discuss the relationship between the approach using roots and this method.

An optimal service policy for buffer systems

Consider a switching component in a packet-switching network, where messages from several incoming channels arrive and are routed to appropriate outgoing ports according to a service policy. One requirement in the design of such a system is to determine the buffer storage necessary at the input of each channel and the policy for serving these buffers that will prevent buffer overflow and the corresponding loss of messages. In this paper, a class of buffer service policies, called Least Time to Reach Bound (LTRB), is introduced that guarantees no overflow, and for which the buffer size required at each input channel is independent of the number of channels and their relative speeds. Further, the storage requirement is only twice the maximal length of a message in all cases, and as a consequence the class is shown to be optimal in the sense that any nonoverflowing policy requires at least as much storage as LTRB.

Matrix-geometric invariant measures for G/M/l type Markov chains

Necessary and sufficient conditions for an irreducible Markov chain of G/M/l type to have an invariant measure that is matrix-geometric are given. For example, it is shown that such an invariant measure exists when a(z), the generating function corresponding to transitions in the homogeneous part of the chain, is either an entire function or a rational function. This generalizes a recent result of Latouche, Pearce and Taylor, who showed that a matrix-geometric invariant measure always exists for level-independent quasi-birth-and-death processes. Conditions ensuring the uniqueness of such an invariant measure up to multiplication by a positive constant are also given. Examples of G/M/l type Markov chains with no matrix-geometric invariant measure and with more than one distinct matrix-geometric invariant measure are presented. As a byproduct of our work, it is shown in the transient case that if det has a solution in the exterior of the closed unit disk, then the solution of smallest modulus there is real ...

Linear independence of root equations for M/G/1 type Markov chains

There is a classical technique for determining the equilibrium probabilities ofM/G/1 type Markov chains. After transforming the equilibrium balance equations of the chain, one obtains an equivalent system of equations in analytic functions to be solved. This method requires finding all singularities of a given matrix function in the unit disk and then using them to obtain a set of linear equations in the finite number of unknown boundary probabilities. The remaining probabilities and other measures of interest are then computed from the boundary probabilities. Under certain technical assumptions, the linear independence of the resulting equations is established by a direct argument involving only elementary results from matrix theory and complex analysis. Simple conditions for the ergodicity and nonergodicity of the chain are also given.

An analysis of a class of telecommunications models

In this short note we outline a general method for characterizing the ergodicity and computing performance measures for a large class of telecommunications models. We also point out errors that have appeared in the literature when using the transform method to analyze such models.

BUFFER SIZE REQUIREMENTS UNDER LONGEST QUEUE FIRST

A model of a switching component in a packet switching network is considered. Packets from several incoming channels arrive and must be routed to the appropriate outgoing port according to a service policy. A task confronting the designer of such a system is the selection of policy and the determination of the corresponding input buffer requirements which will prevent packet loss. One natural choice is the Longest Queue First discipline, and tight bounds on the buffer size required at each channel under this policy are obtained. The bounds depend on the channel speeds and are logarithmic in the number of channels. As a consequence, Longest Queue First is shown to require less storage than Exhaustive Round Robin and First Come First Served in preventing packet overflow.