Several Classes Of Customers Research Articles

Multiclass state‐dependent service systems with returns

AbstractIn this paper, we consider a service system facing several classes of customers in which the arrival rate and service time depend on the workload in the system, while the chance of return is a function of the service time. We first model the problem as a multiclass multiserver queueing network and investigate its stability by examining the conditions under which the Markov chain representing the network is positive recurrent. We then examine the impact of the relationship among the workload, service time, arrival rate, and the chance of return on the dynamics of the system using a fluid approach. We first characterize all equilibria of the system and show that the system may shift between several equilibrium states. We establish that all equilibria can be easily determined and demonstrate conditions under which an equilibrium is stable. We then prove that, surprisingly, the stability of an equilibrium and the congestion in the system may depend on the amount of time a customer spends outside of the system before returning for rework. However, we show that if the relationship between the workload and service time in a system facing a single class of customers is nondecreasing, the long‐run behavior of the system is not affected by how long it takes until a customer returns.

First-come-first-served queues with multiple servers and customer classes

We present a simple approach to the solution of a multi-server FCFS queueing system with several classes of customers and phase-type service time distributions. The proposed solution relies on solving a single two-class model in which we distinguish one of the classes and we aggregate the remaining customer classes. We use a reduced state approximation to solve this two-class model. We propose two types of aggregation: exact, in which we merge the phase-type service time distributions exactly, and approximate, in which we simplify the phase-type distribution for the aggregated class by matching only its first two moments. The proposed approach uses simple mathematics and is highly scalable in terms of the number of servers, the number of classes, as well as the number of phases per class. Our approach applies both to queues with finite and infinite buffer space.

Differentiation service of level in multiservice call centers with adaptation servers

In this article study large-scale service systems with several classes of customers. The following question of the necessary number of servers and compliance with the requirements of customers to reduce the cost of personnel in conditions of limiting the quality of service at the class level is considered. The characteristics of the planning and staffing scheme that are asymptotically optimal under constrained conditions are made, since the load on the system increases to infinity. Asymptotic modes are considered in accordance with the quality and efficiency of management, the quality of management and the effectiveness of management within a single class of service.

Multi-class M/PH/1 queues with deterministic impatience times

ABSTRACTThis article considers computational procedures for the waiting time and queue length distributions in stationary multi-class first-come, first-served single-server queues with deterministic impatience times. There are several classes of customers, which are distinguished by deterministic impatience times (i.e., maximum allowable waiting times). We assume that customers in each class arrive according to an independent Poisson process and a single server serves customers on a first-come, first-served basis. Service times of customers in each class are independent and identically distributed according to a phase-type distribution that may differ for different classes. We first consider the stationary distribution of the virtual waiting time and then derive numerically feasible formulas for the actual waiting time distribution and loss probability. We also analyze the joint queue length distribution and provide an algorithmic procedure for computing the probability mass function of the stationary joint queue length.

Networks of queues models with several classes of customers and exponential service times

The main target of this paper is to present the Markov chain C that, not giving explicitly the queue lengths stationary probabilities, has the necessary information to its determination for open networks of queues with several classes of customers and exponential service times, allowing to overcome ingeniously this problem. The situation for closed networks, in the same conditions, much easier is also presented.

Joint inventory and pricing decisions when customers are delay sensitive

We consider the joint pricing and inventory problem of a capacity constrained service facility with several classes of customers. The customers are differentiated with their sensitivity for waiting and their willingness to pay for the service. We model the problem using an M/M/1 queueing system with non-preemptive priorities. We give closed form solutions for the inventory decisions. We also show that the prices given by the first order conditions are also incentive compatible in the sense that they optimize the profit of the firm even if the firm does not know the type of an arriving customer and let the customer choose a price from the menu of prices. We approximate the problem and provide simple and explicit solutions when there is a single customer type. In numerical illustrations, we show that the customers, who are more sensitive to wait, do not enter the system until the base stock level is above a threshold. We provide extensions of our results for M/G/1 and M/M/m systems.

The fluid limit of the multiclass processor sharing queue

Consider a single server queueing system with several classes of customers, each having its own renewal input process and its own general service times distribution. Upon completing service, customers may leave, or re-enter the queue, possibly as customers of a different class. The server is operating under the egalitarian processor sharing discipline. Building on prior work by Gromoll et al. (Ann. Appl. Probab. 12:797–859, 2002) and Puha et al. (Math. Oper. Res. 31(2):316–350, 2006), we establish the convergence of a properly normalized state process to a fluid limit characterized by a system of algebraic and integral equations. We show the existence of a unique solution to this system of equations, both for a stable and an overloaded queue. We also describe the asymptotic behavior of the trajectories of the fluid limit.

Weak Convergence and Fluid Limits in Optimal Time-to-Empty Queueing Control Problems

We consider a class of controlled queue length processes, in which the control allocates each server’s effort among the several classes of customers requiring its service. Served customers are routed through the network according to (prescribed) routing probabilities. In the fluid rescaling, \(X^{n}(t)=\frac{1}{n} X(nt)\), we consider the optimal control problem of minimizing the integral of an undiscounted positive running cost until the first time that Xn=0. Our main result uses weak convergence ideas to show that the optimal value functions Vn of the stochastic control problems for Xn(t) converge (as n→∞) to the optimal value V of a control problem for the limiting fluid process. This requires certain equicontinuity and boundedness hypotheses on {Vn}. We observe that these are essentially the same hypotheses that would be needed for the Barles-Perthame approach in terms of semicontinuous viscosity solutions. Sufficient conditions for these equicontinuity and boundedness properties are briefly discussed.

Stock rationing in an M/E r /1 multi-class make-to-stock queue with backorders

A model of a single-item make-to-stock production system is presented. The item is demanded by several classes of customers arriving according to Poisson processes with different backorder costs. Item processing times have an Erlang distribution. It is shown that certain structural properties of optimal stock and capacity allocation policies exist for the case where production may be interrupted and restarted. Also, a complete characterization of the optimal policy in the case of uninterrupted production when excess production can be diverted to a salvage market is presented. A heuristic policy is developed and assessed based on the results obtained in the analysis. Finally the value of production status information and the effects of processing time variability are investigated. [Supplemental materials are available for this article. Go to the publisher's online edition of IIE Transactions for the following free supplemental resource: Appendix]

Modeling urban taxi services with multiple user classes and vehicle modes

This paper extends the model of urban taxi services in congested networks to the case of multiple user classes, multiple taxi modes, and customer hierarchical modal choice. There are several classes of customers with different values of time and money, and several modes of taxi services with distinct combinations of service area restrictions and fare levels. The multi-class multi-mode formulation allows the modeling of both mileage-based and congestion-based taxi fare charging mechanisms in a unified framework, and can more realistically model most urban taxi services, which are charged on the basis of both time and distance. The introduction of multiple taxi modes can also be used to model the differentiation between luxury taxis and normal taxis by their respective service areas and customer waiting times. We propose a simultaneous mathematical formulation of two equilibrium sub-problems for the model. One sub-problem is a combined network equilibrium model (CNEM) that describes the hierarchical logit mode choice model of occupied taxis and normal traffic, together with the vacant taxi distributions in the network. The other sub-problem is a set of linear and nonlinear equations (SLNE), which ensures the satisfaction of the relation between taxi and customer waiting times, the relation between customer demand and taxi supply for each taxi mode, and taxi service time constraints. The CNEM can be formulated as a variational inequality program that is solvable by means of a block Gauss–Seidel decomposition approach coupled with the method of successive averages. The SLNE can be solved by a Newtonian algorithm with a line search. The CNEM is formulated as a special case of the general travel demand model so that it is possible to incorporate the taxi model into an existing package as an add-on module, in which the algorithm for the CNEM is built in practice. Most of the parameters are observable, given that such a calibrated transport planning model exists. A numerical example is used to demonstrate the effectiveness of the proposed methodology.

The M/G/1 FIFO queue with several customer classes

In this note we present short derivations of the joint queue length distribution in the M/G/1 queue with several classes of customers and FIFO service discipline.

Optimal Stock Allocation for a Capacitated Supply System

We consider a capacitated supply system that produces a single item that is demanded by several classes of customers. Each customer class may have a different backorder cost, so stock allocation arises as a key decision problem. We model the supply system as a multi customer make-to-stock queue. Using dynamic programming, we show that the optimal allocation policy has a simple and intuitive structure. In addition, we present an efficient algorithm to compute the parameters of this optimal allocation policy. Finally, for a typical supply chain design problem, we illustrate that ignoring the stock allocation dimension—a frequently encountered simplifying assumption—can lead to incorrect managerial decisions.

ANALYSIS OF MULTICLASS M/G/1 QUEUES WITH A MIXTURE OF 1-LIMITED DISCIPLINES AND GATED DISCIPLINES

In this paper, we consider single server queues with several groups and several classes of customers. We consider priority scheduling algorithms for the multiclass queues in which the server admits customers in each group into the service facility by 1-limited disciplines or by gated disciplines. Our objective is to show a method for deriving mean waiting times for these multiclass M/G/l queues. From the analysis of the busy periods, we investigate some linear structure inherent in the mean waiting times conditioned on the system state at each customer's arrival epoch. The steady state mean waiting times can be derived from the linear structure by using the Little's formula and the PASTA property.

A product-form approximation method for general closed queueing networks with several classes of customers

This paper presents a new method for obtaining approximate solutions of general closed queueing networks with several classes of customers. The idea is to associate with each class of customers, a single-class closed queueing network with load-dependent exponential service stations. For each single-class network associated with a particular class of customers, the interactions with the customers belonging to other classes are taken into account in the estimation of the load-dependent service rates. These parameters are obtained by analyzing each station in isolation under the assumption that the arrival process of each class is a state-dependent Markovian process. An iterative algorithm is used to determine the unknown parameters. The main computational charge of the method is due to the analysis of the stations in isolation. A class aggregation technique is proposed that significantly reduces the complexity of these analyses. Although the principle of the method is fairly general, in this paper we only consider queueing networks with general service time distributions and either FIFO or priority disciplines. Numerical results are provided that show that this method is fairly accurate, even when the class aggregation technique is used.

A multi-echelon queueing model with dynamic priority scheduling

In this paper, we consider a multiple finite source queueing model with multiple servers and a nonpreemptive dynamic priority service discipline. The model consists of several classes of customers each of which calls for service at the service facility after spending a random amount of time at the source. The service facility has several servers for serving all customer classes. The service time of customers is exponentially distributed. A nonpreemptive dynamic priority service discipline is used by the servers. In this type of scheduling, a customer's priority function value increases linearly with actual waiting time in the queue. We propose a recursive algorithm that approximates the mean waiting time in the queue of each type of customer classes. Although our solution is only approximate, results from a variety of test problems indicate that the approximation is quite accurate.

Blocking When Service Is Required From Several Facilities Simultaneously

This paper analyzes a mathematical model of a blocking system with simultaneous resource possession. There are several multiserver service facilities without extra waiting space at which several classes of customers arrive in independent Poisson processes. Each customer requests service from one server in each facility in a subset of the service facilities, with the subset depending on the customer class. If service can be provided immediately upon arrival at all required facilities, then service begins and all servers assigned to the customer start and finish together. Otherwise, the attempt is blocked (lost without generating retrials). The problem is to determine the blocking probability for each customer class. An exact expression is available, but it is complicated. Hence, this paper investigates approximation schemes.

The Application of Semi-Markov Decision Processes to Queueing of Aircraft for Landing at an Airport

Socially optimal control of access to the landing queue of an airport is investigated in this paper. Semi-Markov decision process models, M/M/1 and M/Ek/1, for several classes of customers are used to determine how access should be controlled. Only commercial jet aircraft are considered and they are divided up into five classes based on aircraft type. Data from the Greater Pittsburgh International Airport are used to determine the parameters of the service time distribution. Data from several published sources are used to develop the cost, reward, and arrival rate parameters for each of the classes. Results are developed and presented for both models.

Optimal control of entry to an M/Ek/1 queue serving several classes of customers

AbstractThe individual and social optimum control policies for entry to an M/M//1 queue serving several classes of customers have been shown to be control‐limit policies. The technique of policy iteration provides the social optimum policy for such a queue in a straightforward manner. In this article, the problem of finding the optimal control policy for the M/Ek/1 system is solved, thereby expanding the potential applicability of the solutions developed. The Markovian nature of the queueing system is preserved by considering the service as having k sequential phases, each with independent, identically distributed, exponential service times, through which a customer must pass to be serviced. The optimal policy derived by policy iteration for such a system is likely to be difficult to use because it requires knowledge of the number of phases rather than customers in the system when an arrival occurs. To circumvent this difficulty, a heuristic is used to find a good usable (implementable) solution. In addition, a mixed‐integer program is developed which yields the optimal implementable solution when solved.

Exponential spectra as a tool for the study of server-systems with several classes of customers

For a single-server system having several Poisson streams of customers with exponentially distributed service times, busy period densities, waiting time densities, and idle state probabilities are completely monotone. The exponential spectra for such densities are of importance for understanding the transient behavior of such systems. Algorithms are given for the computation of such spectra. Applications to heavy traffic situations and priority systems are also discussed.

Exponential spectra as a tool for the study of server-systems with several classes of customers

For a single-server system having several Poisson streams of customers with exponentially distributed service times, busy period densities, waiting time densities, and idle state probabilities are completely monotone. The exponential spectra for such densities are of importance for understanding the transient behavior of such systems. Algorithms are given for the computation of such spectra. Applications to heavy traffic situations and priority systems are also discussed.