Comparison of two randomized policy M/G/1 queues with second optional service, server breakdown and startup

The problem addressed in this paper is to compare the minimum cost of the two randomized control policies in the M/G/1 queueing system with an unreliable server, a second optional service, and general startup times. All arrived customers demand the first required service, and only some of the arrived customers demand a second optional service. The server needs a startup time before providing the first required service until the system becomes empty. After all customers are served in the queue, the server immediately takes a vacation and the system operates the (T, p)-policy or (p, N)-policy. For those two policies, the expected cost functions are established to determine the joint optimal threshold values of (T, p) and (p, N), respectively. In addition, we obtain the explicit closed form of the joint optimal solutions for those two policies. Based on the minimal cost, we show that the optimal (p, N)-policy indeed outperforms the optimal (T, p)-policy. Numerical examples are also presented for illustrative purposes.     

Dynamic control of a flexible server in an assembly-type queue with setup costs

We consider a queueing system with multiple stations attended by a single flexible server. An order arriving at this system needs to go through each station only once but there is no particular precedence relationship among these stations. One can also think of this system as an assembly system where each station processes a different component of an order and once all the components associated with an order are processed they are assembled instantaneously. A holding cost is charged for keeping the orders in the system and there is a penalty associated with the switches of the server between stations. Our objective is to minimize the long-run average costs by dynamically assigning the server to stations based on the system state. Using sample-path arguments, we provide partial characterizations of the optimal policy and provide sufficient conditions under which a simple state-independent policy that works on one order at a time is optimal. We also propose three simple threshold policies and present a numerical study that provides supporting evidence for the superior performance of our double-threshold heuristic.     

On M/G/1 system under NT policies with breakdowns,startup and closedown

This paper studies the vacation policies of an M/G/1 queueing system with server breakdowns, startup and closedown times, in which the length of the vacation period is controlled either by the number of arrivals during the vacation period, or by a timer. After all the customers are served in the queue exhaustively, the server is shutdown (deactivates) by a closedown time. At the end of the shutdown time, the server immediately takes a vacation and operates two different policies: (i) The server reactivates as soon as the number of arrivals in the queue reaches to a predetermined threshold N or the waiting time of the leading customer reaches T units; and (ii) The server reactivates as soon as the number of arrivals in the queue reaches to a predetermined threshold N or T time units have elapsed since the end of the closedown time. If the timer expires or the number of arrivals exceeds the threshold N, then the server reactivates and requires a startup time before providing the service until the system is empty. If some customers arrive during this closedown time, the service is immediately started without leaving for a vacation and without a startup time. We analyze the system characteristics for each scheme.     

Flexible servers in tandem lines with setup costs

We study the dynamic assignment of flexible servers to stations in the presence of setup costs that are incurred when servers move between stations. The goal is to maximize the long-run average profit. We provide a general problem formulation and some structural results, and then concentrate on tandem lines with two stations, two servers, and a finite buffer between the stations. We investigate how the optimal server assignment policy for such systems depends on the magnitude of the setup costs, as well as on the homogeneity of servers and tasks. More specifically, for systems with either homogeneous servers or homogeneous tasks, small buffer sizes, and constant setup cost, we prove the optimality of “multiple threshold” policies (where servers’ movement between stations depends on both the number of jobs in the system and the locations of the servers) and determine the values of the thresholds. For systems with heterogeneous servers and tasks, small buffers, and constant setup cost, we provide results that partially characterize the optimal server assignment policy. Finally, for systems with larger buffer sizes and various service rate and setup cost configurations, we present structural results for the optimal policy and provide numerical results that strongly support the optimality of multiple threshold policies.     

Optimality of D-Policies for an M/G/1 Queue with a Removable Server

We consider an M/G/1 queue with a removable server. When a customer arrives, the workload becomes known. The cost structure consists of switching costs, running costs, and holding costs per unit time which is a nonnegative nondecreasing right-continuous function of a current workload in the system. We prove an old conjecture that D-policies are optimal for the average cost per unit time criterion. It means that for this criterion there is an optimal policy that either runs the server all the time or switches the server off when the system becomes empty and switches it on when the workload reaches or exceeds some threshold D.     

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.     

Optimality of a Threshold Policy in theM/M/ queue with repeated vacations

Consider anM/M/1 queueing system with server vacations where the server is turned off as soon as the queue gets empty. We assume that the vacation durations form a sequence of i.i.d. random variables with exponential distribution. At the end of a vacation period, the server may either be turned on if the queue is non empty or take another vacation. The following costs are incurred: a holding cost ofh per unit of time and per customer in the system and a fixed cost of each time the server is turned on. We show that there exists a threshold policy that minimizes the long-run average cost criterion. The approach we use was first proposed in Blanc et al. (1990) and enables us to determine explicitly the optimal threshold and the optimal long-run average cost in terms of the model parameters.     

Optimal batch service of a polling system under partial information

We consider the optimal scheduling of an infinite-capacity batch server in aN-node ring queueing network, where the controller observes only the length of the queue at which the server is located. For a cost criterion that includes linear holding costs, fixed dispatching costs, and linear service rewards, we prove optimality and monotonicity of threshold scheduling policies.     

Optimal server assignment in a two-stage tandem queueing system

We study Markovian queueing systems consisting of two stations in tandem. There is a dedicated server in each station and an additional server that can be assigned to any station. Assuming that linear holding costs are incurred by jobs in the system and two servers can collaborate to work on the same job, we determine structural properties of optimal server assignment policies under the discounted and the average cost criteria.     

Equilibrium analysis of the observable queues with balking and delayed repairs

The equilibrium threshold balking strategies are investigated for the fully observable and partially observable single-server queues with server breakdowns and delayed repairs. Upon arriving, the customers decide whether to join or balk the queue based on observation of the queue length and status of the server, along with the consideration of waiting cost and the reward after finishing their service. By using Markov chain approach and system cost analysis, we obtain the stationary distribution of queue size of the queueing systems and provide algorithms in order to identify the equilibrium strategies for the fully and partially observable models. Finally, the equilibrium threshold balking strategies and the equilibrium social benefit for all customers are derived for the fully and partially observable system respectively, both with server breakdowns and delayed repairs.     

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.     

Optimal control of the N policy M/G/1 queueing system with server breakdowns and general startup times

This paper deals with an N policy M/G/1 queueing system with a single removable and unreliable server whose arrivals form a Poisson process. Service times, repair times, and startup times are assumed to be generally distributed. When the queue length reaches N(N ? 1), the server is immediately turned on but is temporarily unavailable to serve the waiting customers. The server needs a startup time before providing service until there are no customers in the system. We analyze various system performance measures and investigate some designated known expected cost function per unit time to determine the optimal threshold N at a minimum cost. Sensitivity analysis is also studied.     

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.     

On the convexity of the two-threshold policy for an M/G/1 queue with vacations

In this note, we consider a single server queueing system with server vacations of two types and a two-threshold policy. Under a cost and revenue structure the long-run average cost function is proven to be convex in the lower threshold for a fixed difference between the two thresholds.     

Optimization and sensitivity analysis of controlling arrivals in the queueing system with single working vacation

This paper analyzes the F-policy M/M/1/K queueing system with working vacation and an exponential startup time. The F-policy deals with the issue of controlling arrivals to a queueing system, and the server requires a startup time before allowing customers to enter the system. For the queueing systems with working vacation, the server can still provide service to customers rather than completely stop the service during a vacation period. The matrix-analytic method is applied to develop the steady-state probabilities, and then obtain several system characteristics. We construct the expected cost function and formulate an optimization problem to find the minimum cost. The direct search method and Quasi-Newton method are implemented to determine the optimal system capacity K, the optimal threshold F and the optimal service rates (μ_B,μ_V) at the minimum cost. A sensitivity analysis is conducted to investigate the effect of changes in the system parameters on the expected cost function. Finally, numerical examples are provided for illustration purpose.     

Threshold start-up control policy for polling systems

A threshold start-up policy is appealing for manufacturing (service) facilities that incur a cost for keeping the machine (server) on, as well as for each restart of the server from its dormant state. Analysis of single product (customer) systems operating under such a policy, also known as the N-policy, has been available for some time. This article develops mathematical analysis for multiproduct systems operating under a cyclic exhaustive or globally gated service regime and a threshold start-up rule. It pays particular attention to modeling switchover (setup) times. The analysis extends/unifies existing literature on polling models by obtaining as special cases, the continuously roving server and patient server polling models on the one hand, and the standard M/G/1 queue with N-policy, on the other hand. We provide a computationally efficient algorithm for finding aggregate performance measures, such as the mean waiting time for each customer type and the mean unfinished work in system. We show that the search for the optimal threshold level can be restricted to a finite set of possibilities. This revised version was published online in June 2006 with corrections to the Cover Date.     

Batch arrival queues with vacations and server setup

We study a single server queueing system whose arrival stream is compound Poisson and service times are generally distributed. Three types of idle period are considered: threshold, multiple vacations, and single vacation. For each model, we assume after the idle period, the server needs a random amount of setup time before serving. We obtain the steady-state distributions of system size and waiting time and expected values of the cycle for each model. We also show that the distributions of system size and waiting time of each model are decomposed into two parts, whose interpretations are provided. As for the threshold model, we propose a method to find the optimal value of threshold to minimize the total expected operating cost.     

Infinite horizon asymptotic average optimality for large-scale parallel server networks

We study infinite-horizon asymptotic average optimality for parallel server networks with multiple classes of jobs and multiple server pools in the Halfin–Whitt regime. Three control formulations are considered: (1) minimizing the queueing and idleness cost, (2) minimizing the queueing cost under constraints on idleness at each server pool, and (3) fairly allocating the idle servers among different server pools. For the third problem, we consider a class of bounded-queue, bounded-state (BQBS) stable networks, in which any moment of the state is bounded by that of the queue only (for both the limiting diffusion and diffusion-scaled state processes). We show that the optimal values for the diffusion-scaled state processes converge to the corresponding values of the ergodic control problems for the limiting diffusion. We present a family of state-dependent Markov balanced saturation policies (BSPs) that stabilize the controlled diffusion-scaled state processes. It is shown that under these policies, the diffusion-scaled state process is exponentially ergodic, provided that at least one class of jobs has a positive abandonment rate. We also establish useful moment bounds, and study the ergodic properties of the diffusion-scaled state processes, which play a crucial role in proving the asymptotic optimality.     

On non-equilibria threshold strategies in ticket queues

In many real-life queueing systems, a customer may balk upon arrival at a queueing system, but other customers become aware of it only at the time the balking customer was to start service. Naturally, the balking is an outcome of the queue length, and the decision is based on a threshold. Yet the inspected queue length contains customers who balked. In this work, we consider a Markovian queue with infinite capacity and with customers that are homogeneous with respect to their cost reward functions. We show that that no threshold strategy can be a Nash equilibrium strategy. Furthermore, we show that for any threshold strategy adopted by all, the individual’s best response is a double threshold strategy. That is, join if and only if one of the following is true: (i) the inspected queue length is smaller than one threshold, or (ii) the inspected queue length is larger than a second threshold. Our model is under the assumption that the response time of the server when he finds out that a customer balked is negligible. We also discuss the validity of the result when the response time is not negligible.