Analysis of a last come first served queueing system with customer abandonment

https://doi.org/10.1016/j.cor.2012.03.009Get rights and content

Abstract

Motivated by manufacturing and service applications, we consider a single class multi-server queueing system working under the LCFS discipline of service. After entering the queue, a customer will wait a random length of time for service to begin. If service has not begun by this time she will abandon and be lost. For the GI/GI/s+M queue, we present some structural results to describe the relation between various performance measures and the scheduling policies. We next consider the LCFS M/M/s+M queue and focus on deriving new results for the virtual waiting time and the sojourn time in the queue (either before service or before abandonment). We provide an exact analysis using Laplace–Stieltjes transforms. We also conduct some numerical analysis to illustrate the impact of customer impatience and the discipline of service on performance.

Introduction

In this paper, we consider queueing models for manufacturing and service systems. We investigate the impact of two important features in practice. The first is the customer limited patience, and the second is the discipline of service. We focus on the last come, first served (LCFS) discipline of service. While the first come, first served (FCFS) policy is the most known in practice due to its fairness, the efficiency of other policies (such as LCFS and random selection (RS)) have also been proved.

Motivation of abandonment modeling in practice: Abandonment, or temporal limitation or also reneging, is an important feature in a wide variety of situations that may be encountered in healthcare applications [4], [15], call centers [7], [14], [11], telecommunication networks [2], [21], manufacturing systems where backlogged orders may be canceled [3], manufacturing systems of perishable goods [17], [12], [5], just to name a few. Models incorporating customer impatience are therefore closer to reality, and lead to more accurate analysis.

For several medical procedures, patients face substantial risk of complication or death when treatment (for example organ transplantation) is delayed. When a queue is formed in such a situation, it is more appropriate to serve the patients according to the urgency of their requirements. When the condition of a patient deteriorates to a certain level, the treatment may become no longer required. In such a case, the patient is removed from the queue without service, i.e., she abandons the queue.

There are many examples of perishable products such as food items, chemicals, pharmaceuticals, but also adhesive materials used for plywood, military ordnance, blood, etc. Karaesmen and Deniz [13] report that, in 2004, 22% of the unsalable costs incurred by distributors of consumer packaged goods were due to expired products, and 5.8% of all components of blood processed for transfusion were outdated. As a result, there is a continued need for understanding such systems and investigating the impact of the finiteness of product lifetimes on production and inventory control decisions. There is a considerable literature related to the modeling of perishable inventory systems using queueing systems with impatient customers. Customer abandonment and product perishing are similar phenomena. A customer whose patience time expires leaves the queue and, likewise, a product made to stock whose lifetime expires is removed from the inventory.

A further example of impatient customers is that of aircrafts in queue for landing. These are willing to wait, but only up to a point. An aircraft may be running out of fuel and so has to be prioritizing to land. In military applications, abandonment is an important feature. For example, enemy's aircrafts or missiles (customers) take a finite time to transit an area where interception is possible, and they escape (abandon) if they are not intercepted (served) within this time. In most call center cases, customers waiting in line are impatient. A customer will wait a given length of time for service to begin. If service has not begun by this time she will abandon and be lost. The importance of modeling abandonments in call centers is emphasized by Garnett et al. [7] and Mandelbaum and Zeltyn [16]. Empirical evidence regarding abandonments in call centers can be found in Brown et al. [6]. We refer the reader to Garnett et al. [7], and references therein for simple models assuming exponential impatience, as we do in this paper.

Much of the queueing literature that incorporates impatience is focused on performance evaluation under FCFS. Another policy of interest is LCFS. In this paper, we analyze a multi-server LCFS queue with customer impatience. We first derive some structural results showing the relation between performance measures and the discipline of service. In particular, we prove for the GI/GI/s+M queue that LCFS minimizes (maximizes) the conditional waiting time in queue, given service (abandonment). Second, we derive new results for the special case of a LCFS M/M/S+M queue, for which we compute the Laplace–Stieltjes transforms for the distributions of the virtual waiting time (of an infinitely impatient customer), and the unconditional waiting time in queue. We find that the expected unconditional waiting times under FCFS and LCFS are identical, whereas their probability distributions are different. We also show some numerical examples to illustrate the impact of the features of impatience and of the LCFS policy on performance.

Motivation of the LCFS policy in practice: In a warehouse where items are stacked upwards, the unit on top is taken to fulfill an order, hence, customers (in this case the units) are served under LCFS. In computer systems, it is also quite common that the server works on jobs that are on the top of the stack. A further application is for broadband communication systems using asynchronous transfer mode. In a multi-access communications system, the LCFS discipline is preferred to FCFS when designing the splitting algorithm with tree structure. In the area of the advanced telephone networks, He and Alfa [10] report that LCFS is more efficient than FCFS in the overload systems. In a general context of service systems, Hassin [8] provides and verbally proves interesting observations on the socially optimality of LCFS. An algebraic proof of this result is given in Hassin and Haviv [9]. Recently for a healthcare application, Su and Zenios [19] also proved that the socially optimal ideal can be attained by merely changing the priority rule to LCFS. They consider a situation where patients form a queue and wait for organs. They observed that patients' quality requirements under FCFS are more stringent than is socially optimal, and their behavior causes excessive organ wastage. Under FCFS, queued patients do not consider the congestion externalities they impose to future arrivals when they decline an organ offer. Inspired by this observation, Su and Zenios [19] consider LCFS, and they demonstrate that patients internalize the externalities of their own decisions, which allows the system performance to achieve the socially optimal ideal.

The remainder of this paper is structured as follows. In Section 2, we describe our model and define various performance measures related to queueing delays. In Section 3, we develop preliminary results that would help us in the rest of the analysis. In Section 4, we develop an approach that allows to characterize the virtual and unconditional waiting times in queue. Some numerical experiments are also shown in order to illustrate the impact of the features of impatience and the discipline of service on performance.

Section snippets

Model description and notations

We consider a multi-server queueing system with a single type of customers. The model consists of one infinite LCFS queue and a set of s parallel, identical servers. The system is work-conserving, i.e., a server is never forced to be idle with customers waiting. Upon arrival, a customer is addressed by one of the available servers, if any. If not, the customer joins the queue. The arrival process is assumed to follow a Poisson process with rate λ. Successive service times are assumed to be

Preliminaries

We start with a tangential development that we will need along the way. Let us relax some assumptions in our original system by considering a GI/GI/s+M queue. We assume that inter-arrival and service times are i.i.d., but we allow them to follow general distributions. Recall that we choose to assign service times to servers and not to arrivals. In Lemma 1, we investigate the relation between the performance measures of interest and the scheduling policies under which queues are working. Within

Analysis of queueing delays

In this section, we derive the Laplace–Stieltjes transforms of the probability distribution functions (pdf) of the busy period duration BP, the virtual waiting time V, and the unconditional waiting time W, which allow to fully characterize the distributions of these random variables.

Upon arrival, a customer is immediately addressed by one of the available servers, if any. If not, she has to join the queue. Then, if her virtual waiting time V ends before her patience threshold T, she will enter

Concluding remarks and future research

In this paper, we considered a queueing system in which customers wait for service for only a limited time and leave system if service has not begun within that time. Practical examples of queueing systems with customer impatience include real-time telecommunication systems, inventory systems with perishable items, and more. Under the LCFS discipline of service, we derived the Laplace–Stieltjes transforms of the pdf of the virtual waiting time and the unconditional waiting time in an M/M/s+M

References (21)

There are more references available in the full text version of this article.

Cited by (14)

  • A discrete-time queue with customers with geometric deadlines

    2015, Performance Evaluation
    Citation Excerpt :

    Understanding such systems and investigating the impact of the finiteness of product lifetimes on production and inventory control decisions is thus clearly necessary in a society in which waste is less and less accepted and in which extra costs (e.g., for cleaning up waste) are more and more avoided. For other examples and situations in which abandonments play an important role, we refer to [21,22]. There is clearly no shortage of continuous-time models to study queues with customer impatience (see, e.g., Zeltyn and Mandelbaum [23] and references therein for a good overview).

  • Implementation of an ant colony approach to solve multi-objective order picking problem in beverage warehousing with drive-in rack system

    2017, 2017 International Conference on Advanced Computer Science and Information Systems, ICACSIS 2017
View all citing articles on Scopus
View full text