D/M/1 Queue: Policies and Control

TL;DR

This paper investigates the equilibrium waiting time distributions in D/M/1 queues under various client-sorting policies, deriving explicit formulas and exploring optimal control of interarrival times based on cost trade-offs.

Contribution

It provides explicit density formulas for D/M/1 queues with LIFO and SIRO policies and analyzes optimal interarrival times considering combined waiting and idling costs.

Findings

Explicit density formulas for D/M/1-LIFO and SIRO queues.

Optimal interarrival time depends on cost trade-offs.

Analysis of how policies affect queue performance.

Abstract

Equilibrium G/M/1-FIFO waiting times are exponentially distributed, as first proved by Smith (1953). For other client-sorting policies, such generality is not feasible. Assume that interarrival times are constant. Symbolics for the D/M/1-LIFO density are completely known; numerics for D/M/1-SIRO arise via an unpublished recursion due to Burke (1967). Consider a weighted sum of two costs, one from keeping clients waiting for treatment and the other from having the server idle. With this in mind, what is the optimal interarrival time and how does this depend on the choice of policy?