Reneging Phenomena in Single Channel Queues

Abstract

We consider a GI/G/1 queueing system where the nth arrival may renege if his service does not commence before an elapsed random time Zn. Results for the general case include relations analogous to Little's formula L = λW, an expression for the average fraction of customers who renege from the system, expressions for the waiting time distribution for all arrivals to the system, the waiting time distribution for arrivals who reach the server, the distribution for virtual waiting time in the queue, and the distribution of the steady-state number of customers in the system and the number of customers left in the system by a completed service. These expressions are written in terms of the distribution of the work seen by an arbitrary arrival to the system, and an integral equation for this distribution is given along with necessary and sufficient conditions for existence of the distribution. Solutions to the integral equation are found for a number of forms for customer interarrival, service, and reneging distributions, and we show how the previous results on the subject of queueing with reneging follow as special cases of our approach. Some past and potential applications of queues with reneging are also discussed.

M/Ph/1 Queue With Deterministic Impatience Time

“…Daley [10] studied the GI/G/1 + G queue by setting up an integral equation for the waiting time distribution and focused on M/G/1 + D and M/G/1 + M (where +M refers to the exponential impatience time) queues. The GI/G/1 + G queue was also studied by Baccelli et al [3] and Stanford [18,19], and results for actual and virtual waiting times were obtained. This paper is inspired by de Kok and Tijms [15] and Xiong et al [21] who studied the M/G/1+D queue.…”

The Busy Period of an M/G/1 Queue with Customer Impatience

“…Stanford [11] relates the waiting time distribution of the (successful) customers and the workload seen by an arbitrary arrival in G/G/1 + G. See Stanford [12] for a brief literature review, and [5] for an approximation for the waiting time distribution in M/G/N + G and several additional references on multiserver queues with impatience.…”

The impact of reneging in processor sharing queues

“…(See also Stanford [22] and Boots and Tijms [6].) This literature focuses on exact performance analysis of the system involved.…”