Transient analysis of a M/M/{\infty} queue with discouragement and for the related embedded chain

TL;DR

This paper analyzes the transient behavior of a discouragement queue modeled as a birth-death process, providing explicit solutions for transition functions and recursive formulas for the embedded chain's distribution.

Contribution

It derives explicit transient solutions for a discouragement queue and formulates recursive expressions for the embedded chain's distribution, extending existing theoretical results.

Findings

Explicit Taylor series solutions for transition functions.

Recursive formulas for the embedded chain's transient distribution.

Enhanced understanding of discouragement queue dynamics.

Abstract

Consider the following birth and death process with the following infinitesimal transition probabilities {\lambda}(k) ={\lambda}/(1+k) and {\mu}(k) = {\mu}k with {\lambda},{\mu}> 0. This process has known as a discouragement queue [5]. Although from the theoretical point of view the problem of the determination of the transition functions has been solved [4], the explicit form of them is not present in literature. We have solved this problem assuming that the solution is representable by a Taylor series, under the initial condition that the process starts to state zero. We discuss also the same problem for the embedded chain and using direct computation we obtain a recursive formula for the transient distribution.