Transient analysis of impatient customers in an M/M/1 disasters queue in random environment

Abstract

Purpose This paper aims to consider a single server queue with system disasters and impatience behavior are evident in our daily life. For this purpose, authors require to know the general behavior of these systems. Transient analysis shows for us how the system will operate up to some time instant t. Design/methodology/approach In this paper, authors consider a single server queue with system disaster and impatient behavior of customers in a multi-phase random environment, in which the system transits to a repair state after each system disaster. When the system is in a failure phase or going through a repair phase, the new arrivals would be impatient. In case the system is not repaired before the customer’s time expires, the customer would leave the queue and never return. Moreover, after repair, the system becomes ready for service in an operative phase with probability $q_{i} \ge 0.$. Using generating functions along with continued fractions and some properties of the confluent hypergeometric function, authors obtained on their own results. Findings Explicit expressions have been obtained for the time-dependent probabilities of the underlying queuing model. Also, time-dependent mean and variance of customers in the system are deduced. Research limitations/implications The system authors are dealing with is somewhat complicated, there are some performance measures that cannot be achieved, but some of them have been obtained, such as the expectation and variance of the number of customers in the system. Practical implications Based on the obtained results, some numerical examples are some numerical examples are presented to illustrate the effect of various parameters on the behavior of the proposed system. Social implications Authors’ studied transient analysis of a single server queue with system disaster and impatient customer system is suitable for behavior interpretation of many systems in our lives, such as telecommunication networks, inventory systems and impatient telephone switchboard customers, manufacturing system and service system. Originality/value To the best of the author’s/authors’ knowledge and according to the literature survey, in a multi-phase random environment, no previous published article is presented for transient analysis of a single server queue with system disaster and impatient customer behavior in a random environment.

Analysis of a Single Server Queueing System Controlled by a Random Switch

“…Several authors have studied single server queueing systems subject to randomly occurring disasters (see, Sengupta [1], Yechiali [2], Chakravarthy [3], Krishna Kumar et al [4], Sudhesh [5], Paz and Yechiali [6], Udayabaskaran and Dora Pravina [7], Kim and Kim [8], Ammar et al [9]). In particular, Paz and Yechiali [6] have analyzed the steady-state behaviour of an M/M/1 queue operating in random environment subject to disasters where the underlying environment is described by a n-level continuous-time Markov chain.…”

“…Udayabaskaran and Dora Pravina [7] have obtained time-dependent probabilities for the queueing model of Paz and Yechiali [6]. Recently, Ammar et al [9] have extended the queueing model of Paz and Yechiali [6] by incorporating customer impatience during repair time, and obtained transient solution for the state probabilities of the system. In the above previous research works, all customers were washed out at the epoch of every disaster.…”

Performance Analysis of A Single Server Queue Operating in A Random Environment - A Novel Approach

“…Queueing systems subject to randomly occurring disasters have been studied by several authors (see, Sengupta [11], Yechiali [15], Chakravarthy [4], Krishna Kumar et al [7], Sudhesh [12], Paz and Yechiali [8], Udayabaskaran and Dora Pravina [13], Kim and Kim [6], Ammar et al [2] ). Queueing systems operating in random environment arise in telecommunication systems and industrial engineering.…”

“…Udayabaskaran and Dora Pravina [13] have obtained time-dependent probabilities for the queueing model of Paz and Yechiali [8]. Recently, Ammar et al [2] have extended the queueing model of Paz and Yechiali [8] by incorporating customer impatience during repair time, and obtained Performance Analysis of A Single Server Queue Operating in A Random Environment -A Novel Approach transient solution for the state probabilities of the system by using generating function technique and continued fraction approach. Queueing systems with set up times for servers have been studied recently by Phung-Duc [9] and Karunakaran and Maragatha Sundari [5].…”

Performance Analysis of Queueing Systems with a Particular Service Interruption Discipline

“…Jain et al [8] studied a fault-tolerant system with general distributed repair time, server vacation, and server breakdown. Ammar et al [9] provided the transient analysis of impatient customers in an M/M/1 disasters queue in random environment. Recently, while the server in a queueing system suffers from unexpected failures and becomes defective, the system can be equipped with a substitute server which continues to provide service for the arriving customers, instead of stopping service completely.…”