Dynamic Routing and Admission Control in High-Volume Service Systems: Asymptotic Analysis via Multi-Scale Fluid Limits

Motivated by applications in telephone call centers, we consider a service system model with m customer classes and r server pools. The model is one with doubly stochastic arrivals, which means that the m-vector λ of instantaneous arrival rates is allowed to vary both temporally and stochastically. Two levels of dynamic control are considered: customers may be either blocked or accepted at the time of their arrival, and then accepted customers of each class must be routed, either immediately upon acceptance or after some period of waiting, to a server pool that is qualified to handle that class. Customers who are made to wait before commencement of their service are liable to defect. The objective is to minimize the expected sum of blocking costs, waiting costs and defection costs over a fixed and finite planning horizon. We consider an asymptotic parameter regime in which (i) the arrival rates, service rates and defection rates are uniformly accelerated by a large factor κ, then (ii) arrival rates are increased by an additional factor g(κ), and the number of servers in each pool is increased by g(κ) as well. This produces a separation of time scales, justifying a pointwise stationary stochastic fluid approximation for our original system model. In the stochastic fluid approximation, optimal admission control and routing decisions are determined by a simple linear program that uses the current arrival rate vector λ as data. We explain how to implement the fluid model's optimal control policy in our original service system context, and prove that the proposed implementation is asymptotically optimal in the first-order sense. AMS subject classification: 60K30, 90B15, 90B36     

Optimal Hysteretic Control for the BMAP/G/ 1 System with Single and Group Service Modes

In this paper, we consider a single server queuing model with an infinite buffer in which customers arrive according to a batch Markovian arrival process (BMAP). The services are offered in two modes. In mode 1, the customers are served one at a time and in mode 2 customers are served in groups of varying sizes. Various costs for holding, service and switching are imposed. For a given hysteretic strategy, we derive an expression for the cost function from which an optimal hysteretic control can be obtained. Illustrative numerical examples are presented.     

Analysis of ticket queues with reneging customers

In this paper we analyse queues in which customer waiting positions are represented by ticket numbers. The customers at any time can observe the number being served and may leave the queue without obtaining the service (reneging). Assuming the customers’ tendency to renege depends dynamically on the difference between their ticket number and the number being served, we develop an approximation procedure in order to calculate the percentage of reneging customers. We give a detailed exposition of the analysis for the case of single-server system and provide a highlight of extension to multi-server systems. As an application of the approximating procedure, we also illustrate numerically that, under a hypothetical reneging behaviour, offering customers extra information on the actual queue length can reduce the customer reneging percentage by as much as 65%.     

Approximate sojourn time distribution of a discriminatory processor sharing queue with impatient customers

We consider a two-class processor sharing queueing system scheduled by the discriminatory processor sharing discipline. Poisson arrivals of customers and exponential amounts of service requirements are assumed. At any moment of being served, a customer can leave the system without completion of its service. In the asymptotic regime, where the ratio of the time scales of the two-class customers is infinite, we obtain the conditional sojourn time distribution of each class customers. Numerical experiments show that the time scale decomposition approach developed in this paper gives a good approximation to the conditional sojourn time distribution as well as the expectation of it.     

Numerical investigation of a multiserver retrial model

We consider a queueing model in which customers arrive in a Poisson stream to be served by one ofc servers. Each arriving customer enters a pool of active customers and starts generating requests for service at exponentially distributed time intervals at rate until he finds a free server and begins service. An analytical solution of this model is difficult and does not lend itself to numerical implementation. In this paper, we make a simplifying approximation, based on understanding of the physical behavior of the system, which yields an infinitesimal generator with a modified matrix-geometric equilibrium probability vector. That vector can be very efficiently computed even for high congestion levels. Illustrative numerical examples demonstrate the effectiveness of the approximation as well as the effect of the retrial rate on the system behavior for various levels of congestion. This study shows how numerical results for analytically intractable systems can be obtained by combining intuition with efficient algorithmic methods.This author's research was supported in part by Grants Nos. ECS-88-03061 from the National Science Foundation and AFOSR-88-0076 from the Air Force Office of Scientific Research.     

A multi-class discrete-time queueing system under the FCFS service discipline

The problem with the FCFS server discipline in discrete-time queueing systems is that it doesn’t actually determine what happens if multiple customers enter the system at the same time, which in the discrete-time paradigm translates into ‘during the same time-slot’. In other words, it doesn’t specify in which order such customers are served. When we consider multiple types of customers, each requiring different service time distributions, the precise order of service even starts to affect quantities such as queue content and delays of arbitrary customers, so specifying this order will be prime. In this paper we study a multi-class discrete-time queueing system with a general independent arrival process and generally distributed service times. The service discipline is FCFS and customers entering during the same time-slot are served in random order. It will be our goal to search for the steady-state distribution of queue content and delays of certain types of customers. If one thinks of the time-slot as a continuous but bounded time period, the random order of service is equivalent to FCFS if different customers have different arrival epochs within this time-slot and if the arrival epochs are independent of customer class. For this reason we propose two distinct ways of analysing; one utilizing permutations, the other considering a slot as a bounded continuous time frame.     

Control and recovery from rare congestion events in a large multi-server system

We develop deterministic fluid approximations to describe the recovery from rare congestion events in a large multi-server system in which customer holding times have a general distribution. There are two cases, depending on whether or not we exploit the age distribution (the distribution of elapsed holding times of customers in service). If we do not exploit the age distribution, then the rare congestion event is a large number of customers present. If we do exploit the age distribution, then the rare event is an unusual age distribution, possibly accompanied by a large number of customers present. As an approximation, we represent the large multi-server system as an M/G/∞ model. We prove that, under regularity conditions, the fluid approximations are asymptotically correct as the arrival rate increases. The fluid approximations show the impact upon the recovery time of the holding-time distribution beyond its mean. The recovery time may or not be affected by the holding-time distribution having a long tail, depending on the precise definition of recovery. The fluid approximations can be used to analyze various overload control schemes, such as reducing the arrival rate or interrupting services in progress. We also establish large deviations principles to show that the two kinds of rare events have the same exponentially small order. We give numerical examples showing the effect of the holding-time distribution and the age distribution, focusing especially on the consequences of long-tail distributions. This revised version was published online in June 2006 with corrections to the Cover Date.     

Exponential Queues with Server Vacations

In this paper, we analyse a queueing system where the server may take a vacation. The customers arrive at the service facility according to a Poisson process, and are served if the server is available (not on vacation). We consider two models: when the server vacation cycle is independent of and dependent on the number of customers in the system. The infinitesimal generators of the underlying Markov processes have a block tri-diagonal structure, and we provide a matrix geometric solution. When the vacation cycle is independent of the customer queue length, we present a simple load-dependent approximation that is fairly accurate.     

Fluid approximations for a processor-sharing queue

In this paper a fluid approximation, also known as a functional strong law of large numbers (FSLLN) for a GI/G/1 queue under a processor-sharing service discipline is established and its properties are analysed. The fluid limit depends on the arrival rate, the service time distribution of the initial customers, and the service time distribution of the arriving customers. This is in contrast to the known result for the GI/G/1 queue under a FIFO service discipline, where the fluid limit is piecewise linear and depends on the service time distribution only through its mean. The piecewise linear form of the limit can be recovered by an equilibrium type choice of the initial service distribution. This revised version was published online in June 2006 with corrections to the Cover Date.     

Analysis of a discrete-time preemptive resume priority buffer

In this paper, we analyze a discrete-time preemptive resume priority queue. We consider two classes of customers which have to be served, where customers of one class have preemptive resume priority over customers of the other. Both classes contain customers with generally distributed service times. We show that the use of probability generating functions is beneficial for analyzing the system contents and customer delays of both classes. It is shown (theoretically as well as by some practical procedures) how moments and approximate tail probabilities of system contents and customer delays are calculated. The influence of the priority scheduling discipline and the service time distributions on the performance measures is shown by some numerical examples.     

Pricing and setup/closedown policies in unobservable queues with strategic customers

In this paper we study unobservable Markovian queueing systems with three types of setup/closedown policies: interruptible, skippable and insusceptible setup/closedown policies, respectively. For a system with the interruptible setup/closedown policy, service starts as soon as a customer arrives during a closedown time; However, for a system with the skippable setup/closedown policy, customers arriving in a closedown time (if any) can be served only after the closedown time finishes and the following setup time can be skipped; Then for a system with the insusceptible setup/closedown policy, customers arriving in a closedown time can??t be served until the following setup time finishes. We assume that customers need a price for service, and derive the equilibrium and socially optimal balking strategies for customers as well as the maximal social welfare. Then we make pricing control to motivate customers to adopt the optimal strategies and obtain an appropriate price that also maximizes server??s profit. Moreover, we numerically make some comparisons between the various performance measures.     

An M/G/1 queue with cyclic service times

In this paper we consider a single server queue with Poisson arrivals and general service distributions in which the service distributions are changed cyclically according to customer sequence number. This model extends a previous study that used cyclic exponential service times to the treatment of general service distributions. First, the stationary probability generating function and the average number of customers in the system are found. Then, a single vacation queueing system with aN-limited service policy, in which the server goes on vacation after servingN consecutive customers is analyzed as a particular case of our model. Also, to increase the flexibility of using theM/G/1 model with cyclic service times in optimization problems, an approximation approach is introduced in order to obtain the average number of customers in the system. Finally, using this approximation, the optimalN-limited service policy for a single vacation queueing system is obtained.On leave from the Department of Industrial Engineering, Iran University of Science and Technology, Narmak, Tehran 16844, Iran.     

Fluid limits of many-server queues with abandonments,general service and continuous patience time distributions

This paper extends the works of Kang and Ramanan (2010) and Kaspi and Ramanan (2011), removing the hypothesis of absolute continuity of the service requirement and patience time distributions. We consider a many-server queueing system in which customers enter service in the order of arrival in a non-idling manner and where reneging is considerate. Similarly to Kang and Ramanan (2010), the dynamics of the system are represented in terms of a process that describes the total number of customers in the system as well as two measure-valued processes that record the age in service of each of the customers being served and the “potential” waiting times. When the number of servers goes to infinity, fluid limit is established for this triple of processes. The convergence is in the sense of probability and the limit is characterized by an integral equation.     

Fluid models of many-server queues with abandonment

We study many-server queues with abandonment in which customers have general service and patience time distributions. The dynamics of the system are modeled using measure-valued processes, to keep track of the residual service and patience times of each customer. Deterministic fluid models are established to provide a first-order approximation for this model. The fluid model solution, which is proved to uniquely exist, serves as the fluid limit of the many-server queue, as the number of servers becomes large. Based on the fluid model solution, first-order approximations for various performance quantities are proposed.     

On the sojourn times for many-queue head-of-the-line Processor-sharing systems with permanent customers

We consider a single server system consisting of e queues with different types of customers (Poisson streams) andk permanent customers. The permanent customers and those at the head of the queues are served in processor-sharing by the service facility (head-of-the-line processor-sharing). The stability condition and a pseudo work conservation law will be given for arbitrary service time distributions; for exponential service times a pseudo conservation law for the mean sojourn tunes can be derived. In case of two queues and exponential service times, the generating function of the stationary occupancy distribution satisfies a functional equation being a Riemann-Hilbert problem which can be reduced to a Dirichlet problem for a circle. The solution yields the mean sojourn times as an elliptic integral, which can be computed numerically very efficiently. In case ofn 2 a numerical algorithm for computing the performance measures is presented, which is efficient forn 3. Since forn 4 an exact analytical or/and numerical treatment is too complex a heuristic approximation for the mean sojourn times of the different types of customers is given, which in case of a (completely) symmetric system is exact. The numerical and simulation results show that, over a wide range of parameters, the approximation works well.This work was supported by a grant from the Siemens AG.     

Multi-server retrial queue with negative customers and disasters

We consider a multi-server retrial queue with waiting places in service area and four types of arrivals, positive customers, disasters and two types of negative customers, one for deleting customers in orbit and the other for deleting customers in service area. The four types of arrivals occur according to a Markovian arrival process with marked transitions (MMAP) which may induce the dependence among the arrival processes of the four types. We derive a necessary and sufficient condition for the system to be positive recurrent by comparing sample paths of auxiliary systems whose stability conditions can be obtained. We use a generalized truncated system that is obtained by modifying the retrial rates for an approximation of stationary queue length distribution and show the convergence of approximation to the original model. An algorithmic solution for the stationary queue length distribution and some numerical results are presented.      

Dynamic load balancing in parallel queueing systems: Stability and optimal control

We consider a system of parallel queues with dedicated arrival streams. At each decision epoch a decision-maker can move customers from one queue to another. The cost for moving customers consists of a fixed cost and a linear, variable cost dependent on the number of customers moved. There are also linear holding costs that may depend on the queue in which customers are stored. Under very mild assumptions, we develop stability (and instability) conditions for this system via a fluid model. Under the assumption of stability, we consider minimizing the long-run average cost. In the case of two-servers the optimal control policy is shown to prefer to store customers in the lowest cost queue. When the inter-arrival and service times are assumed to be exponential, we use a Markov decision process formulation to show that for a fixed number of customers in the system, there exists a level S such that whenever customers are moved from the high cost queue to the low cost queue, the number of customers moved brings the number of customers in the low cost queue to S. These results lead to the development of a heuristic for the model with more than two servers.     

Optimal admission and pricing control problem with deterministic service times and sideline profit

We consider a discrete-time admission control problem in a company operating in service industries with two classes of customers. For the first class of customers, the company then (1) has an option to accept or reject him/her (admission control), or (2) decides on an offering price (pricing control). The second-class (sideline) customers are only served if no first-class customers are in the system, and this yields the sideline profit. In this paper, we discuss both admission control and pricing control problems within an identical framework, and we examine the properties of the optimal policies to maximize the total expected present discounted net profits. We show that when the sideline profit is large, the optimal policies may not be monotone in the number of first-class customers in the system.     

Strong approximations for a Kumar-Seidman network under a priority service discipline

In this paper,we derive the strong approximations for a four-class two station multi-class queuing network（Kumar-Seidman network） under a priority service discipline.Within a group,jobs are served in the order of arrival,that is,a first-in-first-out disciple,and among groups,jobs are served under a pre-emptiveresume priority disciple.We show that if the input data（i.e.,the arrival and service processe） satisfy an approximation（such as the functional law-of-iterated logarithm approximation or the strong approximation）,the output data（the departure processes） and the performance measures（such as the queue length,the work load and the sojourn time process） satisfy a similar approximation.     

Assembly-like queues with finite capacity: Bounds,asymptotics and approximations

In this paper we study a queueing model of assembly-like manufacturing operations. This study was motivated by an examination of a circuit pack testing procedure in an AT & T factory. However, the model may be representative of many manufacturing assembly operations. We assume that customers fromn classes arrive according to independent Poisson processes with the same arrival rate into a single-server queueing station where the service times are exponentially distributed. The service discipline requires that service be rendered simultaneously to a group of customers consisting of exactly one member from each class. The server is idle if there are not enough customers to form a group. There is a separate waiting area for customers belonging to the same class and the size of the waiting area is the same for all classes. Customers who arrive to find the waiting area for their class full, are lost. Performance measures of interest include blocking probability, throughput, mean queue length and mean sojourn time. Since the state space for this queueing system could be large, exact answers for even reasonable values of the parameters may not be easy to obtain. We have therefore focused on two approaches. First, we find upper and lower bounds for the mean sojourn time. From these bounds we obtain the asymptotic solutions as the arrival rate (waiting room, service rate) approaches zero (infinity). Second, for moderate values of these parameters we suggest an approximate solution method. We compare the results of our approximation against simulation results and report good correspondence.