Loss rates in the single-server queue with complete rejection

Bert Zwart

doi:10.1007/s00186-015-0497-x

Abstract

Consider the single-server queue in which customers are rejected if their total sojourn time would exceed a certain level $$K$$ . A basic performance measure of this system is the probability $$P_K$$ that a customer gets rejected in steady state. This paper presents asymptotic expansions for $$P_K$$ as $$K\rightarrow \infty $$ . If the service time $$B$$ is light-tailed and inter-arrival times are exponential, it is shown that the loss probability has an exponential tail. The proof of this result heavily relies on results on the two-sided exit problem for Lévy processes with no positive jumps. For heavy-tailed (subexponential) service times and generally distributed inter-arrival times, the loss probability is shown to be asymptotically equivalent to the trivial lower bound $$P(B>K)$$ .

Highlights

This paper considers the following variation of the single-server queue: customers that arrive are accepted if and only if their total sojourn time is less than a fixed constant K
We are interested in the probability PK that a customer is rejected in steady state, more precisely, in the behavior of PK as K → ∞
We develop a description of the workload process in the single-server queue with complete rejection

Summary

Introduction

This paper considers the following variation of the single-server queue: customers that arrive are accepted if and only if their total sojourn time is less than a fixed constant K. The following result (which holds for the G I /G/1 queue with partial rejection) can be found in Bekker and Zwart (2005): PKp = P(Vmax > K ) Another tractable model is the M/G/1 queue where customers leave the system due to impatience when their waiting time has exceeded a fixed threshold K. With WM/G/1 the steady-state waiting time distribution in the M/G/1 queue, see Boots and Tijms (1999) These formulas can be applied to obtain asymptotic expansions for PKp or PKi , since the asymptotic behavior of P(WM/G/1 > K ) and P(Vmax > K ) is well known for both the light-tailed and the heavy-tailed case.

The single-server queue with complete rejection

The single-server queue with infinite buffer size

Main results

The two-sided exit problem

Other rejection mechanisms