Optimal buffer size and dynamic rate control for a queueing system with impatient customers in heavy traffic

We address a rate control problem associated with a single server Markovian queueing system with customer abandonment in heavy traffic. The controller can choose a buffer size for the queueing system and also can dynamically control the service rate (equivalently the arrival rate) depending on the current state of the system. An infinite horizon cost minimization problem is considered here. The cost function includes a penalty for each rejected customer, a control cost related to the adjustment of the service rate and a penalty for each abandoning customer. We obtain an explicit optimal strategy for the limiting diffusion control problem (the Brownian control problem or BCP) which consists of a threshold-type optimal rejection process and a feedback-type optimal drift control. This solution is then used to construct an asymptotically optimal control policy, i.e. an optimal buffer size and an optimal service rate for the queueing system in heavy traffic. The properties of generalized regulator maps and weak convergence techniques are employed to prove the asymptotic optimality of this policy. In addition, we identify the parameter regimes where the infinite buffer size is optimal.     

Dynamic admission and service rate control of a queue

This paper investigates a queueing system in which the controller can perform admission and service rate control. In particular, we examine a single-server queueing system with Poisson arrivals and exponentially distributed services with adjustable rates. At each decision epoch the controller may adjust the service rate. Also, the controller can reject incoming customers as they arrive. The objective is to minimize long-run average costs which include: a holding cost, which is a non-decreasing function of the number of jobs in the system; a service rate cost c(x), representing the cost per unit time for servicing jobs at rate x; and a rejection cost κ for rejecting a single job. From basic principles, we derive a simple, efficient algorithm for computing the optimal policy. Our algorithm also provides an easily computable bound on the optimality gap at every step. Finally, we demonstrate that, in the class of stationary policies, deterministic stationary policies are optimal for this problem.     

Optimal pricing for tandem queues with finite buffers

We consider optimal pricing for a two-station tandem queueing system with finite buffers, communication blocking, and price-sensitive customers whose arrivals form a homogeneous Poisson process. The service provider quotes prices to incoming customers using either a static or dynamic pricing scheme. There may also be a holding cost for each customer in the system. The objective is to maximize either the discounted profit over an infinite planning horizon or the long-run average profit of the provider. We show that there exists an optimal dynamic policy that exhibits a monotone structure, in which the quoted price is non-decreasing in the queue length at either station and is non-increasing if a customer moves from station 1 to 2, for both the discounted and long-run average problems under certain conditions on the holding costs. We then focus on the long-run average problem and show that the optimal static policy performs as well as the optimal dynamic policy when the buffer size at station 1 becomes large, there are no holding costs, and the arrival rate is either small or large. We learn from numerical results that for systems with small arrival rates and no holding cost, the optimal static policy produces a gain quite close to the optimal gain even when the buffer at station 1 is small. On the other hand, for systems with arrival rates that are not small, there are cases where the optimal dynamic policy performs much better than the optimal static policy.

     

The effect of different arrival rates on the N-policy of M/G/1 with server setup

It is very important in many real-life systems to decide when the server should start his service because frequent setups inevitably make the operating cost too high. Furthermore, today's systems are too intelligent for the input to be assumed as a simple homogenous Poisson process. In this paper, an M/G/1 queue with general server setup time under a control policy is studied. We consider the case when the arrival rate varies according to the server's status: idle, setup and busy states. We derive the distribution function of the steady-state queue length, as well as the Laplace–Stieltjes transform of waiting time. For this model, the optimal N-value from which the server starts his setup is found by minimizing the total operation cost of the system. We finally investigate the behavior of system operation cost and the optimal N for various arrival rates by a numerical study.     

Convexity properties of the overflow in an ordered-entry system with heterogeneous servers

The overflow probability in an Erlang loss system is known to be decreasing convex in the number of servers. Here we consider the GI/M/m loss system with ordered entry and heterogeneous servers. We show that adding a sequence of servers with non-increasing (non-decreasing) service rates will yield a decreasing convex (log-concave) sequence of overflow probabilities. An optimal server allocation problem is solved as a direct application of these results.     

Modelling complex assemblies as a queueing network for lead time control

In this paper we develop an open queueing network for optimal design of multi-stage assemblies, in which each service station represents a manufacturing or assembly operation. The arrival processes of the individual parts of the product are independent Poisson processes with equal rates. In each service station, there is a server with exponential distribution of processing time, in which the service rate is controllable. The transport times between the service stations are independent random variables with exponential distributions. By applying the longest path analysis in queueing networks, we obtain the distribution function of time spend by a product in the system or the manufacturing lead time. Then, we develop a multi-objective optimal control problem, in which the average lead time, the variance of the lead time and the total operating costs of the system per period are minimized. Finally, we use the goal attainment method to obtain the optimal service rates or the control vector of the problem.     

Optimal Release Times in Single-Stage Manufacturing Systems with Blocking: Optimal Control Perspective

This paper concerns a due-date matching problem in a single-stage manufacturing system. Given a finite sequence of jobs and their service order, and given the delivery due date of each job, the problem is to choose the jobs release (arrival) times so as to match as closely as possible their completion times to their respective due dates. The system is modelled as a deterministic single-server FIFO queue with an output buffer for storing jobs whose service is completed prior to their due dates. The output buffer has a finite capacity; when it is full, the server is being blocked. Associated with each job there is a convex cost function penalizing its earliness as well as tardiness. The due-date matching problem is cast as an optimal control problem, whose objective is to minimize the sum of the above cost functions by the choice of the jobs arrival (release) times. Time-box upper-bound and lower-bound constraints are imposed on the jobs output (delivery) times. The optimal-control setting brings to bear on the development of fast and efficient algorithms having intuitive geometric appeal and potential for online implementation.Communicated by W. B. GongResearch supported in part by the National Science Foundation under Grant ECS-9979693 and by the Georgia Tech Manufacturing Research Center under Grant B01-D06.     

An optimal service rate in a Poisson arrival queue with two-stage service policy*

We consider a two-stage service policy for a Poisson arrival queueing system. The idle server starts to work with ordinary service rate when a customer arrives. If the number of customers in the system reaches N, the service rate gets faster and continues until the system becomes empty. Otherwise, the server finishes the busy period with ordinary service rate. After assigning various operating costs to the system, we show that there exists a unique fast service rate minimizing the long-run average cost per unit time.This work was supported by Korea Research Foundation Grant(KRF-2002-070-C00021).     

Service rate control of closed Jackson networks from game theoretic perspective

Game theoretic analysis of queueing systems is an important research direction of queueing theory. In this paper, we study the service rate control problem of closed Jackson networks from a game theoretic perspective. The payoff function consists of a holding cost and an operating cost. Each server optimizes its service rate control strategy to maximize its own average payoff. We formulate this problem as a non-cooperative stochastic game with multiple players. By utilizing the problem structure of closed Jackson networks, we derive a difference equation which quantifies the performance difference under any two different strategies. We prove that no matter what strategies the other servers adopt, the best response of a server is to choose its service rates on the boundary. Thus, we can limit the search of equilibrium strategy profiles from a multidimensional continuous polyhedron to the set of its vertex. We further develop an iterative algorithm to find the Nash equilibrium. Moreover, we derive the social optimum of this problem, which is compared with the equilibrium using the price of anarchy. The bounds of the price of anarchy of this problem are also obtained. Finally, simulation experiments are conducted to demonstrate the main idea of this paper.     

Index Heuristics for Multiclass M/G/1 Systems with Nonpreemptive Service and Convex Holding Costs

We consider the optimal service control of a multiclass M/G/1 queueing system in which customers are served nonpreemptively and the system cost rate is additive across classes and increasing convex in the numbers present in each class. Following Whittle's approach to a class of restless bandit problems, we develop a Langrangian relaxation of the service control problem which serves to motivate the development of a class of index heuristics. The index for a particular customer class is characterised as a fair charge for service of that class. The paper develops these indices and reports an extensive numerical investigation which exhibits strong performance of the index heuristics for both discounted and average costs.     

Dynamic job assignment: A column generation approach with an application to surgery allocation

We consider the assignment of jobs to heterogeneous agents in a dynamic system with a rolling time horizon. An example is a hospital operating theatre where the jobs are surgeries and the agents are the surgeons. The paper is presented in the context of surgery allocation and the system is characterized as follows: Patients are grouped into categories and they arrive continually following a stochastic process. Patients in each group have specific time limits within which they need treatment and if it cannot be accommodated then the patients are outsourced. The service level is the percentage of patients in each group treated within the time limit. Surgery durations are stochastic and depend on the surgeon conducting the surgeries. Each surgeon has limited time available and expected overtime is penalized by a non-decreasing convex function. We develop a column generation approach for the assignment of already arrived patients and tentative future patients to surgeons on specific days. It balances the conflicting objectives of including as many arrived patients as possible within their time limits, maximizing the service level of future patients, and minimizing the expected overtime of surgeons. A computational study is conducted with the model embedded in a rolling time horizon frame. The study indicates that the assignment of patients based on our model increases system performance in terms of service level and reduced overtime compared to a First-Come-First-Served (FCFS) policy when the arrival rates of patients are medium to high compared to the capacity of the system.     

Time-cost trade-off via optimal control theory in Markov PERT networks

We develop a new analytical model for the time-cost trade-off problem via optimal control theory in Markov PERT networks. It is assumed that the activity durations are independent random variables with generalized Erlang distributions, in which the mean duration of each activity is a non-increasing function of the amount of resource allocated to it. Then, we construct a multi-objective optimal control problem, in which the first objective is the minimization of the total direct costs of the project, in which the direct cost of each activity is a non-decreasing function of the resources allocated to it, the second objective is the minimization of the mean of project completion time and the third objective is the minimization of the variance of project completion time. Finally, two multi-objective decision techniques, viz, goal attainment and goal programming are applied to solve this multi-objective optimal control problem and obtain the optimal resources allocated to the activities or the control vector of the problem     

Optimal design of multi‐server Markovian queues with polynomial waiting and service costs

This paper is concerned with the optimal design of queueing systems. The main decisions in the design of such systems are the number of servers, the appropriate control to have on the arrival rates, and the appropriate service rate these servers should possess. In the formulation of the objective function to this problem, most publications use only linear cost rates. The linear rates, especially for the waiting cost, do not accurately reflect reality. Although there are papers involving nonlinear cost functions, no paper has ever considered using polynomial cost functions of degree higher than two. This is because simple formulas for computing the higher moments are not available in the literature. This paper is an attempt to fill this gap in the literature. Thus, the main contributions of our work are as follows: (i) the derivation of a very simple formula for the higher moments of the waiting time for the M/M/s queueing system, which requires only the knowledge of the expected waiting time; (ii) proving their convexity with respect to the design variables; and (iii) modeling and solving more realistic design problems involving general polynomial cost functions. We also focus on simultaneous optimization of the staffing level, arrival rate and service rate. Copyright © 2013 John Wiley & Sons, Ltd.     

Dynamic Scheduling of a Multiclass Fluid Model with Transient Overload

We study the optimal dynamic scheduling of different requests of service in a multiclass stochastic fluid model that is motivated by recent and emerging computing paradigms for Internet services and applications. In particular, our focus is on environments with specific performance guarantees for each class under a profit model in which revenues are gained when performance guarantees are satisfied and penalties are incurred otherwise. Within the context of the corresponding fluid model, we investigate the dynamic scheduling of different classes of service under conditions where the workload of certain classes may be overloaded for a transient period of time. Specifically, we consider the case with two fluid classes and a single server whose capacity can be shared arbitrarily among the two classes. We assume that the class 1 arrival rate varies with time and the class 1 fluid can more efficiently reduce the holding cost. Under these assumptions, we characterize the optimal server allocation policy that minimizes the holding cost in the fluid model when the arrival rate function for class 1 is known. Using the insights gained from this deterministic case, we study the stochastic fluid system when the arrival rate function for class 1 is random and develop various policies that are optimal or near optimal under various conditions. In particular, we consider two different types of heavy traffic regimes and prove that our proposed policies are strongly asymptotically optimal. Numerical examples are also provided to demonstrate further that these policies yield good results in terms of minimizing the expected holding cost.     

Inventory control with two switch-over levels for a class of M/G/1 queueing systems with variable arrival and service rate

This paper deals with inventory control in a class of M/G/1 queueing systems. At each point of time the system can be switched from one of two possible stages to another. The rate of arrival process and the service rate depend on the stage of the system. The cost structure imposed on the model includes both fixed switch-over costs and a holding cost at a general rate depending on the stage of the system. The rule for controlling the inventory is specified by two switch-over levels.Using an embedding approach, we will derive a formula for the long-run average expected costs per unit time of this policy. By an appropriate choice of the cost parameters, we may obtain various operating characteristics for the system amongst which the stationary distribution of the inventory and the average number of switch-overs per unit time.     

A search method for optimal control of a flow shop system of traditional machines

We consider a convex and nondifferentiable optimization problem for deterministic flow shop systems in which the arrival times of the jobs are known and jobs are processed in the order they arrive. The decision variables are the service times that are to be set only once before processing the first job, and cannot be altered between processes. The cost objective is the sum of regular costs on job completion times and service costs inversely proportional to the controllable service times. A finite set of subproblems, which can be solved by trust-region methods, are defined and their solutions are related to the optimal solution of the optimization problem under consideration. Exploiting these relationships, we introduce a two-phase search method which converges in a finite number of iterations. A numerical study is held to demonstrate the solution performance of the search method compared to a subgradient method proposed in earlier work.     

M/G/1 queue with event-dependent arrival rates

Motivated by experiments on customers’ behavior in service systems, we consider a queueing model with event-dependent arrival rates. Customers’ arrival rates depend on the last event, which may either be a service departure or an arrival. We derive explicitly the performance measures and analyze the impact of the event-dependency. In particular, we show that this queueing model, in which a service completion generates a higher arrival rate than an arrival, performs better than a system in which customers are insensitive to the last event. Moreover, contrary to the M/G/1 queue, we show that the coefficient of variation of the service does not necessarily deteriorate the system performance. Next, we show that this queueing model may be the result of customers’ strategic behavior when only the last event is known. Finally, we investigate the historical admission control problem. We show that, under certain conditions, a deterministic policy with two thresholds may be optimal. This new policy is easy to implement and provides an improvement compared to the classical one-threshold policy.     

Control of arrivals in a finite buffered queue with setup costs

We consider finite buffered queues where the arrival process is controlled by shutting down and restarting the arrival stream. In the absence of holding costs for items in the queue, the optimal (s,?S) policy can be characterised by relating the arrival control problem to a corresponding service control problem. With the inclusion of holding costs however, this characterisation is not valid and efficient numerical computation of the queue length probability distribution is necessary. We perform this computation by using a duality property which relates queue lengths in the controlled arrival system to a controlled service system. Numerical results which analyse the effect of setup and holding costs and the variability of the arrival process on the performance of the system are included.     

An inventory model for deteriorating items with stock-dependent consumption rate and shortages under inflation and time discounting

This paper derives an inventory model for deteriorating items with stock-dependent consumption rate and shortages under inflation and time discounting over a finite planning horizon. We show that the total cost function is convex. With the convexity, a simple solution algorithm is presented to determine the optimal order quantity and the optimal interval of the total cost function. The results are discussed with a numerical example and particular cases of the model are discussed in brief. A sensitivity analysis of the optimal solution with respect to the parameters of the system is carried out.     

Multiproduct single-machine production system with stochastic scrapped production rate, partial backordering and service level constraint

In this paper, a multiproduct single-machine production system under economic production quantity (EPQ) model is studied in which the existence of only one machine causes a limited production capacity for the common cycle length of all products, the production defective rates are random variables, shortages are allowed and take a combination of backorder and lost sale, and there is a service rate constraint for the company. The aim of this research is to determine the optimal production quantity, the allowable shortage level, and the period length of each product such that the expected total cost, including holding, shortage, production, setup and defective items costs, is minimized. The mathematical model of the problem is derived for which the objective function is proved to be convex. Then, a derivative approach is utilized to obtain the optimal solution. Finally, two numerical examples in each of which a sensitivity analysis is performed on the model parameters, are provided to illustrate the practical usage of the proposed methodology.