In this talk, we discuss equilibrium solutions of a multi-server queue with two or more unbounded state variables and propose a method to approximate these equilibrium solutions by transient solutions. By doing so, we aim to provide an algorithm for decision-making at any time during the operation of a queueing system. Events in discrete systems tend to increase or decrease state variables by fixed amounts. If an increase is possible in all states, then there is no natural boundary. For instance, in certain queueing systems, one might disallow arrivals once the queue length reaches a certain limit to keep all queues finite. However, removing such bounds is desirable for many applications. Most standard queueing models (e.g., M/M/c, GI/M/c, M/G/1, Jackson networks, and tandem queues) allow for queues of infinite length. Our goal is to handle systems that have no limit on two or more physical state variables.

For the simpler case with only one unbounded state variable, an extension of Crout’s method—called the extrapolating Crout method—was introduced. Another group of well-known techniques for solving systems with one unbounded state variable are matrix-analytic methods (MAM), which can be derived from the extrapolating Crout method. Crout’s approach is often easier to understand and can be faster than traditional MAM-based methods. We now consider the case with two unbounded state variables. We start with simpler models leading to block-structured transition matrices, where rows and columns repeat beyond certain boundaries.

Accordingly, we obtain repeated block matrices of unbounded sizes. Let the blocks be denoted by Pk for −g ≤ k ≤ h, where g and h are fixed positive integers. For a simple case, let X1 and X2 be two unbounded state variables. We aim to determine the equilibrium vectors πi. Thus, if πi,j is the equilibrium probability that X1 = i and X2 = j, the desired vector is πi = [πi,j , j = 1, 2, . . .], i = 1, 2, . . .