Analysis of customers’ impatience in queues with server vacations

Abstract

Many models for customers impatience in queueing systems have been studied in the past; the source of impatience has always been taken to be either a long wait already experienced at a queue, or a long wait anticipated by a customer upon arrival. In this paper we consider systems with servers vacations where customers' impatience is due to an absentee of servers upon arrival. Such a model, representing frequent behavior by waiting customers in service systems, has never been treated before in the literature. We present a comprehensive analysis of the single-server, M/M/1 and M/G/1 queues, as well as of the multi-server M/M/c queue, for both the multiple and the single-vacation cases, and obtain various closed-form results. In particular, we show that the proportion of customer abandonments under the single-vacation regime is smaller than that under the multiple-vacation discipline. Keywords

Synchronized reneging in queueing systems with vacations

“…The simplest case is the one where all servers take a vacation when the system becomes empty and all of them return as in the oneserver case. This agrees to the many-server case with independent abandonments in the paper of Altman and Yechiali (2006). We can also consider the single-vacation case which is different from the multiple-vacation case described above in that the server takes just one vacation and then remains to the system even if there are not waiting customers.…”

“…Using these facts we can study the case of no synchronization (i.e., independent abandonments) that has been investigated by Altman and Yechiali (2006). The following theorem corresponds to their results for the M/M/1 type model (see their Sect.…”

“…At the abandonment epochs, every present customer remains in the system with probability q or abandons the system with probability p = 1 − q, independently of the others. The analysis of this model extends the analysis of Altman and Yechiali (2006), in the framework of synchronized abandonments. The new feature of these models with synchronization is the existence of binomial type jumps at the abandonment epochs.…”

“…Although, there exists a significant number of papers and books on vacation queueing systems (see, e.g., Takagi (1991) and Tian and Zhang (2006)), the reneging feature has not yet received much attention. Only recently, Altman and Yechiali (2006) and Yechiali (2007) considered systems with vacations, where the source of the impatience is the absence of the server. The authors assume that the customers perform independent abandonments, whenever the server is unavailable.…”

Control of G/MX/1 Queueing System with N-Policy and Customer Impatience

“…Blackburn (1972) considers an M/G/1 queueing system with customer balking and/or reneging and gives a finite algorithm to compute the stationary optimal policy that maximizes the expected discounted reward under some conditions on the system parameters and unit costs. Altman and Yechiali (2006) presents a comprehensive analysis of the M/M/c and M/G/1 queues, where customers' impatience is due to the absence of servers upon arrival for both the multiple and the singlevacation cases, and obtain various closed-form results. More recently, Yue et al (2011) consider a two-phase queueing system with impatient customers and multiple vacations with Poisson arrivals.…”