# The time-dependent expected reward and deviation matrix of a finite QBD process

## TL;DR

This paper develops methods to compute the time-dependent expected reward and deviation matrix for finite QBD processes, aiding in performance analysis of queueing systems like MAP/PH/1/C.

## Contribution

It introduces two approaches—matrix difference equations and perturbation theory—for calculating the deviation matrix of finite QBD processes, with practical numerical comparisons.

## Key findings

- Both methods effectively compute the deviation matrix.
- The recursive approach offers computational advantages.
- Numerical examples demonstrate the methods' applicability.

## Abstract

Deriving the time-dependent expected reward function associated with a continuous-time Markov chain involves the computation of its transient deviation matrix. In this paper we focus on the special case of a finite quasi-birth-and-death (QBD) process, motivated by the desire to compute the expected revenue lost in a MAP/PH/1/C queue. We use two different approaches in this context. The first is based on the solution of a finite system of matrix difference equations; it provides an expression for the blocks of the expected reward vector, the deviation matrix, and the mean first passage time matrix. The second approach, based on some results in the perturbation theory of Markov chains, leads to a recursive method to compute the full deviation matrix of a finite QBD process. We compare the two approaches using some numerical examples.