Stochastic scheduling of parallel queues with set-up costs

We consider the problem of allocating a single server to a system of queues with Poisson arrivals. Each queue represents a class of jobs and possesses a holding cost rate, general service distribution, and a set-up cost. The objective is to minimize the expected cost due to the waiting of jobs and the switching of the server. A set-up cost is required to effect an instantaneous switch from one queue to another. We partially characterize an optimal policy and provide a simple heuristic scheduling policy. The heuristic's performance is evaluated in the cases of two and three queues by comparison with a numerically obtained optimal policy. Simulation results are provided to demonstrate the effectiveness of our heuristic over a wide range of problem instances with four queues.     

Optimal control of polling models for transportation applications

We formulate and analyze a dynamic scheduling problem for a class of transportation systems in a Markov Decision Process (MDP) framework. A transportation system is represented by a polling model consisting of a number of stations and a server with switch-over costs and constraints on its movement (the model we have analyzed is intended to emulate key features of an elevator system). Customers request service in order to be transported by the server from various arrival stations to a common destination station. The objective is to minimize a cost criterion that incorporates waiting costs at the arrival stations. Two versions of the basic problem are considered and structural properties of the optimal policy in each case are derived. It is shown that optimal scheduling policies are characterized by switching functions dependent on state information consisting of queue lengths formed at the arrival stations.     

An optimal maintenance policy for a server with decreasing arrival rate

We study an optimal maintenance policy for the server in a queueing system. Customers arrive at the server in a Poisson stream and are served by an exponential server, which is subject to multiple states indicating levels of popularity. The server state transitions are governed by a Markov process. The arrival rate depends on the server state and it decreases as the server loses popularity. By maintenance the server state recovers completely, though the customers in the system are lost at the beginning of maintenance. The customers who arrive during maintenance are also lost. In this paper, two kinds of such systems are considered. The first system receives a unit reward when a customer arrives at the system and pays a unit cost for each lost customer at the start of maintenance. The second system receives a unit reward at departure, and pays nothing for lost customers at the beginning of maintenance. Our objective is to maximize the total expected discounted profit over an infinite time horizon. We use a semi-Markov decision process to formulate the problem and are able to establish some properties for the optimal maintenance policy under certain conditions.     

Optimal control of parallel server systems with many servers in heavy traffic

We consider a parallel server system that consists of several customer classes and server pools in parallel. We propose a simple robust control policy to minimize the total linear holding and reneging costs. We show that this policy is asymptotically optimal under the many-server heavy traffic regime for parallel server systems when the service times are only server pool dependent and exponentially distributed. J.G. Dai’s research supported in part by National Science Foundation grants CMMI-0727400 and CNS-0718701, and by an IBM Faculty Award.     

A fuzzy based threshold policy for a single server retrial queue with vacations

In this paper we deal with a single server retrial queue with vacations. The server serves the customers until the system becomes empty, then it takes a vacation. The system consists of two types of costs. The blocking cost is considered whenever a customer is blocked either because of the server is busy or off. There is also a cost each time the server is turned on. The problem is to find an effective policy for turning on the dormant server. We propose a Fuzzy Based Threshold Policy (FBTP) to control the server, substitute for conventional threshold policies. The FBTP is based on four input parameters, an inference stage and it is tuned up using a stochastic List Based Threshold Accepting (LBTA) algorithm. Simulation models are developed to validate the fuzzy controller. Numerical experiments are provided to show that the proposed method is superior to crisp threshold policies.     

On priority queues with impatient customers

In this paper, we study three different problems where one class of customers is given priority over the other class. In the first problem, a single server receives two classes of customers with general service time requirements and follows a preemptive-resume policy between them. Both classes are impatient and abandon the system if their wait time is longer than their exponentially distributed patience limits. In the second model, the low-priority class is assumed to be patient and the single server chooses the next customer to serve according to a non-preemptive priority policy in favor of the impatient customers. The third problem involves a multi-server system that can be used to analyze a call center offering a call-back option to its impatient customers. Here, customers requesting to be called back are considered to be the low-priority class. We obtain the steady-state performance measures of each class in the first two problems and those of the high-priority class in the third problem by exploiting the level crossing method. We furthermore adapt an algorithm from the literature to obtain the factorial moments of the low-priority queue length of the multi-server system exactly.      

On optimal polling policies

In a single-server polling system, the server visits the queues according to a routing policy and while at a queue, serves some or all of the customers there according to a service policy. A polling (or scheduling) policy is a sequence of decisions on whether to serve a customer, idle the server, or switch the server to another queue. The goal of this paper is to find polling policies that stochastically minimize the unfinished work and the number of customers in the system at all times. This optimization problem is decomposed into three subproblems: determine the optimal action (i.e., serve, switch, idle) when the server is at a nonempty queue; determine the optimal action (i.e., switch, idle) when the server empties a queue; determine the optimal routing (i.e., choice of the queue) when the server decides to switch. Under fairly general assumptions, we show for the first subproblem that optimal policies are greedy and exhaustive, i.e., the server should neither idle nor switch when it is at a nonempty queue. For the second subproblem, we prove that in symmetric polling systems patient policies are optimal, i.e., the server should stay idling at the last visited queue whenever the system is empty. When the system is slotted, we further prove that non-idling and impatient policies are optimal. For the third subproblem, we establish that in symmetric polling systems optimal policies belong to the class of Stochastically Largest Queue (SLQ) policies. An SLQ policy is one that never routes the server to a queue known to have a queue length that is stochastically smaller than that of another queue. This result implies, in particular, that the policy that routes the server to the queue with the largest queue length is optimal when all queue lengths are known and that the cyclic routing policy is optimal in the case that the only information available is the previous decisions.This work was supported in part by NSF under Contract ASC-8802764.     

An analysis of Klimov's problem with parallel servers

We introduce a family of undiscounted branching bandits on parallel servers under controls which can impose priorities between customer classes. This family can be used to model a wide range of multi-class queueing scheduling problems, with the capacity to incorporate problem features such as machine breakdowns, complex patterns of preemption/non-preemption and semi-Markov extensions. An index policy (which we call Klimov's rule) is developed which is optimal in the particular case of a single server. An expression for its cost suboptimality is given for parallel servers. Under additional conditions on the nature of the stochastic evolution of the systems concerned, the policy is shown to be asymptotically optimal in a heavy traffic limit. These general results are utilised to develop an analysis of the index policy for a parallel server version of Klimov's classical M/GI/1 system with Bernoulli feedback. This work was supported by the Engineering and Physical Research Council through the award of grant GR/M09308. The author would also like to express his appreciation to Professor I. C. Paschalidis for helpful discussions on Klimov's problem and to Professors J. Niño-Mora and G. Weiss for many discussions and much encouragement     

Scheduling networks of queues: Heavy traffic analysis of a simple open network

We consider a queueing network with two single-server stations and two types of customers. Customers of type A require service only at station 1 and customers of type B require service first at station 1 and then at station 2. Each server has a different general service time distribution, and each customer type has a different general interarrival time distribution. The problem is to find a dynamic sequencing policy at station 1 that minimizes the long-run average expected number of customers in the system.The scheduling problem is approximated by a dynamic control problem involving Brownian motion. A reformulation of this control problem is solved, and the solution is interpreted in terms of the queueing system in order to obtain an effective sequencing policy. Also, a pathwise lower bound (for any sequencing policy) is obtained for the total number of customers in the network. We show via simulation that the relative difference between the performance of the proposed policy and the pathwise lower bound becomes small as the load on the network is increased toward the heavy traffic limit.     

Dynamic scheduling of a single-server two-class queue with constant retrial policy

We analyze the non-preemptive assignment of a single server to two infinite-capacity retrial queues. Customers arrive at both queues according to Poisson processes. They are served on first-come-first-served basis following a cost-optimal routing policy. The customer at the head of a queue generates a Poisson stream of repeated requests for service, that is, we have a constant retrial policy. All service times are exponential, with rates depending on the queues. The costs to be minimized consist of costs per unit time that a customer spends in the system. In case of a scheduling problem that arise when no new customers arrive an explicit condition for server allocation to the first or the second queue is given. The condition presented covers all possible parameter choices. For scheduling that also considers new arrivals, we present the conditions under which server assignment to either queue 1 or queue 2 is cost-optimal.     

A simple policy for multiple queues with size-independent service times

We consider a service system with two Poisson arrival queues. A server chooses which queue to serve at each moment. Once a queue is served, all the customers will be served within a fixed amount of time. This model is useful in studying airport shuttling or certain online computing systems. We propose a simple yet optimal state-independent policy for this problem which is not only easy to implement, but also performs very well.     

Admission control and pricing in a queue with batch arrivals

We investigate a problem of admission control in a queue with batch arrivals. We consider a single server with exponential service times and a compound Poisson arrival process. Each arriving batch computes its expected benefit and decides whether or not to enter the system. The controller’s problem is to set state dependent prices for arriving batches. Once prices have been set we formulate the admission control problem, derive properties of the value function, and obtain the optimal admission policy.     

Control and scheduling in a two-station queueing network: Optimal policies and heuristics

Consider a two-station queueing network with two types of jobs: type 1 jobs visit station 1 only, while type 2 jobs visit both stations in sequence. Each station has a single server. Arrival and service processes are modeled as counting processes with controllable stochastic intensities. The problem is to control the arrival and service processes, and in particular to schedule the server in station 1 among the two job types, in order to minimize a discounted cost function over an infinite time horizon. Using a stochastic intensity control approach, we establish the optimality of a specific stationary policy, and show that its value function satisfies certain properties, which lead to a switching-curve structure. We further classify the problem into six parametric cases. Based on the structural properties of the stationary policy, we establish the optimality of some simple priority rules for three of the six cases, and develop heuristic policies for the other three cases.     

A modified vacation model M[x]/G/1 system

This paper studies the operating characteristics of an M^[x]/G/1 queueing system under a modified vacation policy, where the server leaves for a vacation as soon as the system is empty. The server takes at most J vacations repeatedly until at least one customer is found waiting in the queue when the server returns from a vacation. We derive the system size distribution at different points in time, as well as the waiting time distribution in the queue. Further, we derive some important characteristics including the expected length of the busy period and idle period. This shows that the results generalize those of the multiple vacation policy and the single vacation policy M^[x]/G/1 queueing system. Finally, a cost model is developed to determine the optimum of J at a minimum cost. Copyright © 2006 John Wiley & Sons, Ltd.     

Optimal Operating Policy for an M/G/1 Exhaustive Server-Vacation Model

We consider an M/G/1 queueing system controlled by an exhaustive server–vacation policy, i.e, the server is turned off whenever the system becomes empty and it is turned on after a random time with at least a customer present in the system. In this paper, it is proved that there exists an exhaustive optimal policy which is of the form X + a(T - X)⁺, where, starting with the server off, X represents the time for the first arrival and T and a are non-negative real numbers. Using a classical average cost structure, the optimization problem is treated under the asymptotic average criterion. A structured definition of exhaustive policy is also derived.     

The single-server scheduling problem with convex costs

Being probably one of the oldest decision problems in queuing theory, the single-server scheduling problem continues to be a challenging one. The original formulations considered linear costs, and the resulting policy is puzzling in many ways. The main one is that, either for preemptive or nonpreemptive problems, it results in a priority ordering of the different classes of customers being served that is insensitive to the individual load each class imposes on the server and insensitive to the overall load the server experiences. This policy is known as the cμ-rule. We claim and show that for convex costs, the optimal policy depends on the individual loads. Therefore, there is a need for an alternative generalization of the cμ-rule. The main feature of our generalization consists on first-order differences of the single stage cost function, rather than on its derivatives. The resulting policy is able to reach near optimal performances and is a function of the individual loads.     

Optimal control of a single server in a finite-population queueing network

We study the optimal dynamic assignment of a single server to multiple stations in a finite-population queueing network. The objective is to maximize the long-run average reward/throughput. We use sample-path comparisons to identify conditions on the network structure and service time distributions under which the optimal policy is an index policy. This index policy assigns the server to the non-empty station where it takes the shortest amount of time (in some stochastic sense) to complete a job. For example, in a network of multiple parallel stations, the optimal policy assigns the highest priority to the fastest station if service times can be ordered in likelihood ratios. Finally, by means of a numerical study, we test the shortest-expected-remaining-service-time policy on parallel-series networks with three stations and find that this index policy either coincides with the optimal policy or provides a near-optimal performance.     

Dynamic scheduling to minimize lost sales subject to set-up costs

We consider scheduling a shared server in a two-class, make-to-stock, closed queueing network. We include server switching costs and lost sales costs (equivalently, server starvation penalties) for lost jobs. If the switching costs are zero, the optimal policy has a monotonic threshold type of switching curve provided that the service times are identical. For completely symmetric systems without set-ups, it is optimal to serve the longer queue. Using simple analytical models as approximations, we derive a heuristic scheduling policy. Numerical results demonstrate the effectiveness of our heuristic, which is typically within 10% of optimal. We also develop and test a heuristic policy for a model in which the shared resource is part of a series network under a CONWIP release policy. This revised version was published online in June 2006 with corrections to the Cover Date.     

A server backup model with Markovian arrivals and phase type services

We consider a finite capacity queueing system with one main server who is supported by a backup server. We assume Markovian arrivals, phase type services, and a threshold-type server backup policy with two pre-determined lower and upper thresholds. A request for a backup server is made whenever the buffer size (number of customers in the queue) hits the upper threshold and the backup server is released from the system when the buffer size drops to the lower threshold or fewer at a service completion of the backup server. The request time for the backup server is assumed to be exponentially distributed. For this queuing model we perform the steady state analysis and derive a number of performance measures. We show that the busy periods of the main and backup servers, the waiting times in the queue and in the system, are of phase type. We develop a cost model to obtain the optimal threshold values and study the impact of fixed and variable costs for the backup server on the optimal server backup decisions. We show that the impact of standard deviations of the interarrival and service time distributions on the server backup decisions is quite different for small and large values of the arrival rates. In addition, the pattern of use of the backup server is very different when the arrivals are positively correlated compared to mutually independent arrivals.