MMAP|M|N queueing system with impatient heterogeneous customers as a model of a contact center

A multi-server queueing system with infinite buffer and impatient heterogeneous customers as a model of a contact center that processes incoming calls (priority customers) and e-mail requests (non-priority customers) is investigated. The arrival flow is described by a Marked Markovian Arrival Process (MMAP). The service time of priority and non-priority customers by a server has an exponential distribution with different parameters. The steady state distribution of the system is analyzed. Some key performance measures are calculated. The Laplace–Stieltjes transforms of the sojourn and waiting time distribution are derived. The problem of optimal choice of the number of contact center agents under the constraint that the average waiting time of e-mail requests does not exceed a predefined value is numerically solved.     

Analysis of discrete-time batch service renewal input queue with multiple working vacations

This paper investigates a discrete-time single server batch service queue with multiple working vacations wherein arrivals occur according to a discrete-time renewal process. The server works with a different service rate rather than completely stopping during the vacation period. The service is performed in batches and the server takes a vacation when the system does not have any waiting customers at a service completion epoch or a vacation completion epoch. We present a recursive method, using the supplementary variable technique to obtain the steady-state queue-length distributions at pre-arrival, arbitrary and outside observer’s observation epochs. The displacement operator method is used to solve simultaneous non-homogeneous difference equations. Some performance measures and waiting-time distribution in the system have also been discussed. Finally, numerical results showing the effect of model parameters on key performance measures are presented.     

Queueing analysis and optimal control of and systems

We first consider a finite-buffer single server queue where arrivals occur according to batch Markovian arrival process (BMAP). The server serves customers in batches of maximum size ‘b’ with a minimum threshold size ‘a’. The service time of each batch follows general distribution independent of each other as well as the arrival process. We obtain queue length distributions at various epochs such as, pre-arrival, arbitrary, departure, etc. Some important performance measures, like mean queue length, mean waiting time, probability of blocking, etc. have been obtained. Total expected cost function per unit time is also derived to determine the optimal value N^* of N at a minimum cost for given values of a and b. Secondly, we consider a finite-buffer single server queue where arrivals occur according to BMAP and service process in this case follows a non-renewal one, namely, Markovian service process (MSP). Server serves customers according to general bulk service rule as described above. We derive queue length distributions and important performance measures as above. Such queueing systems find applications in the performance analysis of communication, manufacturing and transportation systems.     

A perishable inventory system with service facilities and retrial customers

In this article, we present a continuous review perishable (s, S) inventory system with a service facility consisting of finite waiting room and a single server. The customers arrive according to a Markovian arrival process (MAP). The individual customer’s unit demand is satisfied after a random time of service which is assumed to have phase-type distribution. The life time of each item and the lead time of reorders are assumed to have independent exponential distributions. Any arriving customer, who finds the waiting room is full, enters into the orbit of infinite space. These orbiting customers compete for service by sending out signals the duration between two successive attempts are exponentially distributed. The joint probability distribution of the number of customers in the waiting room, number of customers in the orbit and the inventory level is obtained in the steady-state case. Various stationary system performance measures are computed and total expected cost rate is calculated.     

Performance analysis of an M/G/1 queueing system under Bernoulli vacation schedules with server setup and close down periods

This paper is concerned with the analysis of a single server queueing system subject to Bernoulli vacation schedules with server setup and close down periods. An explicit expression for the probability generating function of the number of customers present in the system is obtained by using imbedded Markov chain technique. The steady state probabilities of no customer in the system at the end of vacation termination epoch and a service completion epoch are derived. The mean number of customers served during a service period and the mean number of customers in the system at an arbitrary epoch are investigated under steady state. Further, the Laplace-Stieltjes transform of the waiting time distribution and its corresponding mean are studied. Numerical results are provided to illustrate the effect of system parameters on the performance measures.     

A two-moment approximation for a buffer design problem requiring a small rejection probability

A useful model for buffer capacity design in communication systems is the single server queueing model with restricted accessibility where arriving customers are admitted only if their waiting plus service times do not exceed some fixed amount. A two-moment approximation for the buffer capacity in order to achieve a specific rejection probability is proposed for the case of Poisson arrivals and general service requirements. This approximation is a weighted combination of exact results for the special cases of deterministic and exponential service requirements where the weights use only the coefficient of variation of the general service requirement. Numerical experiments show an excellent performance of the approximation.     

On a batch arrival Poisson queue with a random setup time and vacation period

The single server queue with vacation has been extended to include several types of extensions and generalisations, to which attention has been paid by several researchers (e.g. see Doshi, B. T., Single server queues with vacations — a servey. Queueing Systems, 1986, 1, 29–66; Takagi, H., Queueing Analysis: A Foundation of Performance evaluation, Vol. 1, Vacation and Priority systems, Part. 1. North Holland, Amsterdam, 1991; Medhi, J., Extensions and generalizations of the classical single server queueing system with Poisson input. J. Ass. Sci. Soc., 1994, 36, 35–41, etc.). The interest in such types of queues have been further enhanced in resent years because of their theoretical structures as well as their application in many real life situations such as computer, telecommunication, airline scheduling as well as production/inventory systems. This paper concerns the model building of such a production/inventory system, where machine undergoes extra operation (such as machine repair, preventive maintenance, gearing up machinery, etc.) before the processing of raw material is to be started. To be realistic, we also assume that raw materials arrive in batch. This production system can be formulated as an M^x/M/1 queues with a setup time. Further, from the utility point of view of idle time this model can also be formulated as a case of multiple vacation model, where vacation begins at the end of each busy period. Besides, the production/inventory systems, such a model is generally fitted to airline scheduling problems also. In this paper an attempt has been made to study the steady state behavior of such an M^x/M/1 queueing system with a view to provide some system performance measures, which lead to remarkable simplification when solving other similar types of queueing models.This paper deals with the steady state behaviour of a single server batch arrival Poisson queue with a random setup time and a vacation period. The service of the first customer in each busy period is preceded by a random setup period, on completion of which service starts. As soon as the system becomes empty the server goes on vacation for a random length of time. On return from vacation, if he finds customer(s) waiting, the server starts servicing the first customer in the queue. Otherwise it takes another vacation and so on. We study the steady state behaviour of the queue size distribution at random (stationary) point of time as well as at departure point of time and try to show that departure point queue size distribution can be decomposed into three independent random variables, one of which is the queue size of the standard M^x/M/1 queue. The interpretation of the other two random variables will also be provided. Further, we derive analytically explicit expressions for the system state (number of customers in the system) probabilities and provide their appropriate interpretations. Also, we derive some system performance measures. Finally, we develop a procedure to find mean waiting time of an arbitrary customer.     

Performance analysis of a non-preemptive priority queuing system subjected to a correlated Markovian interruption process

In this paper, we consider a discrete-time queuing system with head-of-line non-preemptive priority scheduling and a single server subjected to server interruptions. We model the server interruptions by a correlated Markovian on/off process with geometrically distributed on and off periods. Two classes of traffic are considered, namely high-priority and low-priority traffic. In the first part of the paper, we derive an expression for the functional equation describing the transient evolution of this priority queuing system. This functional equation is then manipulated and transformed into a mathematical tractable form. This allows us to derive the joint probability generating function (pgf) of the system contents. From this pgf, closed-form expressions for various performance measures, such as mean and variance of system contents and customer delay can be derived. Finally, we illustrate our solution technique with some numerical examples, whereby we demonstrate the negative effect of correlation in the interruption process on the performance of both classes. Some numerical results illustrating the impact of second-order characteristics of the arrival process on mean delays are also presented. The proposed approach which is purely based on pgfs is entirely analytical and enables the derivation of not only steady-state but transient performance measures, as well. The paper presents new insights into the performance analysis of discrete-time queues with service interruption and it also covers some previously published results as a special case.

Scope and purpose

In this contribution, we consider a practical queuing model, with HOL priority scheduling, two classes of traffic, and a server which is subjected to a correlated Markovian interruption process. We first derive a non-linear functional equation relating the joint pgf of the system state vector between two consecutive slots. Then we outline a solution technique to solve for this functional equation. This allows us to derive the joint pgf of the system contents of both classes, from which various performance measures related to mean system contents and customer delays are derived. We also demonstrate how the proposed approach allows for derivation of transient performance measures, as well. It should be noted that detailed coverage of the transient analysis of the system is beyond the scope of this paper.To our best knowledge, this is the first initiative that aims to explore the performance of queuing systems with priority scheduling when the shared server is subjected to service interruption. The paper also generalizes the results of Walraevens et al. (Analysis of a single-server ATM queue with priority scheduling, Computers & Operations Research 2003;30(12):1807–30) by incorporating service interruption into their original queuing model. By means of numerical results, the paper also demonstrates the effect of correlation in the service interruption process on the performance of both classes of customers. The impact of second-order characteristics of the arrival process on mean delays is also investigated.     

A finite source multi-server inventory system with service facility

In this article, we study a continuous review retrial inventory system with a finite source of customers and identical multiple servers in parallel. The customers arrive according a quasi-random process. The customers demand unit item and the demanded items are delivered after performing some service the duration of which is distributed as exponential. The ordering policy is according to (s, S) policy. The lead times for the orders are assumed to have independent and identical exponential distributions. The arriving customer who finds all servers are busy or all items are in service, joins an orbit. These orbiting customer competes for service by sending out signals at random times until she finds a free server and at least one item is not in the service. The inter-retrial times are exponentially distributed with parameter depending on the number of customers in the orbit. The joint probability distribution of the number of customer in the orbit, the number of busy servers and the inventory level is obtained in the steady state case. The Laplace–Stieltjes transform of the waiting time distribution and the moments of the waiting time distribution are calculated. Various measures of stationary system performance are computed and the total expected cost per unit time is calculated. The results are illustrated numerically.     

The combined gated-exhaustive vacation system in discrete time

We consider a discrete-time gated vacation system. The available buffer space is divided into two subsequent queues separated by a gate and new customers arrive either before or after this gate. Whenever all customers after the gate are served, the server takes a vacation. After each vacation, the gate opens which causes all waiting customers to move to the buffer space after the gate. The model under investigation allows to capture performance of a.o. the exhaustive and the gated queueing systems with multiple or single vacations. Using a probability generating functions approach, we obtain expressions for performance measures such as moments of system contents at various epochs in equilibrium and of customer delay. We conclude with a numerical example.     

A finite source multi-server inventory system with service facility

In this article, we study a continuous review retrial inventory system with a finite source of customers and identical multiple servers in parallel. The customers arrive according a quasi-random process. The customers demand unit item and the demanded items are delivered after performing some service the duration of which is distributed as exponential. The ordering policy is according to (s, S) policy. The lead times for the orders are assumed to have independent and identical exponential distributions. The arriving customer who finds all servers are busy or all items are in service, joins an orbit. These orbiting customer competes for service by sending out signals at random times until she finds a free server and at least one item is not in the service. The inter-retrial times are exponentially distributed with parameter depending on the number of customers in the orbit. The joint probability distribution of the number of customer in the orbit, the number of busy servers and the inventory level is obtained in the steady state case. The Laplace–Stieltjes transform of the waiting time distribution and the moments of the waiting time distribution are calculated. Various measures of stationary system performance are computed and the total expected cost per unit time is calculated. The results are illustrated numerically.     

Scheduling of multi-class multi-server queueing systems with abandonments

Many real-world situations involve queueing systems in which customers may abandon if service does not start sufficiently quickly. We study a comprehensive model of multi-class queue scheduling accounting for customer abandonment, with the objective of minimizing the total discounted or time-average sum of linear waiting costs, completion rewards, and abandonment penalties of customers in the system. We assume the service times and abandoning times are exponentially distributed. We solve analytically the case in which there is one server and there are one or two customers in the system and obtain an optimal policy. For the general case, we use the framework of restless bandits to analytically design a novel simple index rule with a natural interpretation. We show that the proposed rule achieves near-optimal or asymptotically optimal performance both in single- and multi-server cases, both in overload and underload regimes, and both in idling and non-idling systems.     

Reducing congestion in bulk-service finite-buffer queueing system using batch-size-dependent service

In the design and analysis of any queueing system, one of the main objectives is to reduce congestion which can be achieved by controlling either arrival-rates or service-rates. This paper adopts the latter approach and analyzes a single-server finite-buffer queue where customers arrive according to the Poisson process and are served in batches of minimum size a with a maximum threshold limit b. The service times of the batches are arbitrarily distributed and depends on the size of the batches undergoing service. We obtain the joint distribution of the number of customers in the queue and the number with the server, and distributions of the number of customers in the queue, in the system, and the number with the server. Various performance measures such as the average number of customers in the queue (system) and with the server etc. are obtained. Several numerical results are presented in the form of tables and graphs and it is observed that batch-size-dependent service rule is more effective in reducing the congestion as compared to the one when service rates of the batches remain same irrespective of the size of the batch. This model has potential application in manufacturing, computer-communication network, telecommunication systems and group testing.     

A batch arrival queue under randomised multi-vacation policy with unreliable server and repair

This article examines an M^[x]/G/1 queueing system with an unreliable server and a repair, in which the server operates a randomised vacation policy with multiple available vacations. Upon the system being found to be empty, the server immediately takes a vacation. If there is at least one customer found waiting in the queue upon returning from a vacation, the server will be activated for service. Otherwise, if no customers are waiting for service at the end of a vacation, the server either remains idle with probability p or leaves for another vacation with probability 1???p. When one or more customers arrive when the server is idle, the server immediately starts providing service for the arrivals. It is possible that an unpredictable breakdown may occur in the server, in which case a repair time is requested. For such a system, we derive the distributions of several important system characteristics, such as the system size distribution at a random epoch and at a departure epoch, the system size distribution at the busy period initiation epoch, and the distribution of the idle and busy periods. We perform a numerical analysis for changes in the system characteristics, along with changes in specific values of the system parameters. A cost effectiveness maximisation model is constructed to show the benefits of such a queueing system.     

A repairable queueing model with two-phase service,start-up times and retrial customers

A repairable queueing model with a two-phase service in succession, provided by a single server, is investigated. Customers arrive in a single ordinary queue and after the completion of the first phase service, either proceed to the second phase or join a retrial box from where they retry, after a random amount of time and independently of the other customers in orbit, to find a position for service in the second phase. Moreover, the server is subject to breakdowns and repairs in both phases, while a start-up time is needed in order to start serving a retrial customer. When the server, upon a service or a repair completion finds no customers waiting to be served, he departs for a single vacation of an arbitrarily distributed length. The arrival process is assumed to be Poisson and all service and repair times are arbitrarily distributed. For such a system the stability conditions and steady state analysis are investigated. Numerical results are finally obtained and used to investigate system performance.     

An approximation for mean waiting times in cyclic server systems with nonexhaustive service

The cyclic server system has been the subject of considerable research over the last few years. Interest in analyzing such systems has gained momentum due to their application in the performance analysis of token ring networks. In this paper we consider cyclic server systems with nonexhaustive service discipline. The performance measures of interest here are the mean waiting times at the nodes in the system. Exact analysis of such systems for these performance measures is very difficult in general, and a number of approximation schemes have been proposed in the past to evaluate these quantities. This paper presents a new approximation technique that gives accurate estimates of these mean waiting times, based on extensive validation with simulations.     

Single server retrial queue with group admission of customers

We consider a retrial queueing system with a single server and novel customer׳s admission discipline. The input flow is described by a Markov Arrival Process. If an arriving customer meets the server providing the service, it goes to the orbit and repeats attempts to get service in random time intervals whose duration has exponential distribution with parameter dependent on the customers number in orbit. Server operates as follows. After a service completion epoch, customers admission interval starts. Duration of this interval has phase type distribution. During this interval, primary customers and customers from the orbit are accepted to the pool of customers which will get service after the admission interval. Capacity of this pool is limited and after the moment when the pool becomes full before completion of admission interval all arriving customers move to the orbit. After completion of an admission interval, all customers in the pool are served simultaneously by the server during the time having phase type distribution depending on the customers number in the pool. Using results known for Asymptotically Quasi-Toeplitz Markov Chains, we derive stability condition of the system, compute the stationary distribution of the system states, derive formulas for the main performance measures and numerically show advantages of the considered customer׳s admission discipline (higher throughput, smaller average number of customers in the system, higher probability to get a service without visiting the orbit) in case of proper choice of the capacity of the pool and the admission period duration.     

A queue with working breakdowns

In this paper, we consider a new class of queueing models with working breakdowns. The system may become defective at any point of time when it is in operation. However, when the system is defective, instead of stopping service completely, the service continues at a slower rate. Using the probability generating function, we give the joint distribution of the server state and the number of customers in the system in steady state. We also derive the necessary and sufficient condition for the existence of the steady state. We study the waiting time distribution of our model. Finally, some performance measures and numerical examples are presented.     

The effect of workers with different capabilities on customer delay

Many service facilities operate seven days per week. The operations managers of these facilities face the problem of allocating personnel of varying skills and work speed to satisfy the demand for services. Furthermore, for practical reasons, periodic staffing schedule is implemented regularly. We introduce a novel approach for modeling periodic staffing schedule and analyzing the impact of employee variability on customer delay. The problem is formulated as a multiple server vacation queueing system with Bernoulli feedback of customers. At any point in time, at most one server can serve the customers. Each server incur a durations of set-up time before they can serve the customers. The customer service time and server set-up time may depend on the server. The service process is unreliable in the sense that it is possible for the customer at service completion to rejoin the queue and request for more service. The customer arrival process is assumed to satisfy a linear–quadratic model of uncertainty. We will present transient and steady-state analysis on the queueing model. The transient analysis provides a stability condition for the system to reach steady state. The steady-state analysis provides explicit expressions for several performance measures of the system. For the special case of M^X/G/1 vacation queue with a gated or exhaustive service policy and Bernoulli feedback, our result reduces to a previously known result. Lastly, we show that a variant of our periodic staffing schedule model can be used to analyze queues with permanent customers. For the special case of M/G/1 queue with permanent customers and Bernoulli feedback of ordinary customers, we obtain results previously given by Boxma and Cohen (IEEE J. Select. Areas Commun. 9 (1991) 179) and van den Berg (Sojourn Times in Feedback and Processor Sharing Queues, CWI Tracts, vol. 97, Amsterdam, Netherlands, 1993).Scope and purposeWorkforce scheduling is a classical problem and has been studied by many researchers. The problem is usually formulated with homogeneous workforce as part of the assumption. Clearly, non-homogeneous workforce is a fact of life for many organizations. Operations manager would prefer to have skills and experience worker as it would improve the quality of the services provided. Ignoring the effect of employees with varying skills and work speed would seriously undermine the effectiveness of the services provided and lead to significant undesirable outcomes for the organization. This paper aims as a first step to fill the gap of past research. We present a novel approach to analyze the issue of non-homogeneous workforce on stability of work flows and the effect of workers with different capabilities on customers’ waiting time. We believe that the results are useful for operations manager dealing with non-homogeneous workforce.     

A polynomial factorization approach to the discrete time GI/G/1/(N) queue size distribution

The queue of a single server is considered with independent and identically distributed interarrivai and service times and an infinite (GI/G/1) or finite (GI/G/1/N) waiting room. The queue discipline is non-preemptive and independent of the service times.

A discrete time version of the system is analyzed, using a two-component state model at the arrival and departure instants of customers. The equilibrium equations are solved by a polynomial factorization method. The steady state distribution of the queue size is then represented as a linear combination of geometrical series, whose parameters are evaluated by closed formulae depending on the roots of a characteristic polynomial.

Considering modified boundary constraints, systems with finite waiting room or with an exceptional first service in each busy period are included.