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 use of excess capacity in two interconnected queues

We consider a two-stage tandem queueing network where jobs from station 1 join station 2 with a certain probability. Each job incurs a linear holding cost, different for each station. Each station is attended by a dedicated server, and there is an additional server that is either constrained to serve in station 1 or can serve in both stations. Assuming no switching or other operating costs for the additional server, we seek an allocation strategy that minimizes expected holding costs. For a clearing system we show that the optimal policy is characterized by a switching curve for which we provide a lower bound on its slope. We also specify a subset of the state space where the optimal policy can be explicitly determined.     

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.     

Beyond the cμ rule: Dynamic scheduling of a two-class loss queue

We consider scheduling a single server in a two-class M/M/1 queueing system with finite buffers subject to holding costs and rejection costs for rejected jobs. We use dynamic programming to investigate the structural properties of optimal policies. Provided that the delay of serving a job is always less costly than rejecting an arrival, we show that the optimal policy has a monotonic threshold type of switching curve; otherwise, numerical analysis indicates that the threshold structure may not be optimal. Received December 1996/Revised version May 1997     

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.     

Scheduling a repairman in a finite source system

The scheduling of a single server in a finite source model is considered. TheN customers in the system have different failure and repair rates. Also the costs depend on the customers which are broken down. We give a condition under which the average costs are minimized by a simple list policy, and with a counterexample we show that in the general case no optimal list policy may exist. This motivates us to derive policies which are optimal under low and high traffic conditions. They are again list policies, which behave well numerically.     

Optimal assignment policy of a single server attended by two queues

A single server attends to two separate queues. Each queue has Poisson arrivals and exponential service. There is a switching cost whenever the server switches from one queue to another. The objective is to minimize the discounted or average holding and switching costs over a finite or an infinite horizon. We show numerically that the optimal assignment policy is characterized by a switching curve. We also show that the optimal policy is monotonic in the following senses: If it is optimal to switch from queue one to queue two, then it is optimal to continue serve queue two whenever the number of customers in queue one or in queue two decreases or increases, respectively.     

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.     

Allocating flexible servers in serial systems with switching costs

Consider a firm that operates a make-to-order serial production system and employs a cross-trained workforce. We model such a firm as a tandem queuing system in which flexible servers can be allocated across stations, and assume that a switching cost is charged when servers move between stations. We show that even in the two-station two-server case the optimal policy follows a complex state-dependent structure that may be difficult to implement in practice. We propose three alternate heuristic policies and assess their performance. We show that a simpler policy which only moves one server can achieve close to optimal results.     

A two-stage tandem queue attended by a moving server with holding and switching costs

We consider a two-stage tandem queue attended by a moving server, with homogeneous Poisson arrivals and general service times. Two different holding costs for stages 1 and 2 and different switching costs from one stage to the other are considered. We show that the optimal policy in the second stage is greedy; and if the holding cost rate in the second stage is greater or equal to the rate in the first stage, then the optimal policy in the second stage is also exhaustive. Then, the optimality condition for sequential service policy in systems with zero switchover times is introduced. Considering some properties of the optimal policy, we then define a Triple-Threshold (TT) policy to approximate the optimal policy in the first stage. Finally, a model is introduced to find the optimal TT policy, and using numerical results, it is shown that the TT policy accurately approximates the optimal policy. This revised version was published online in June 2006 with corrections to the Cover Date.     

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.     

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.     

Admission policies for the customized stochastic lot scheduling problem with strict due-dates

This papers considers admission control and scheduling of customer orders in a production system that produces different items on a single machine. Customer orders drive the production and belong to product families, and have family dependent due-date, size, and reward. When production changes from one family to another a setup time is incurred. Moreover, if an order cannot be accepted, it is considered lost upon arrival. The problem is to find a policy that accepts/rejects and schedules orders such that long run profit is maximized. This problem finds its motivation in batch industries in which suppliers have to realize high machine utilization while delivery times should be short and reliable and the production environment is subject to long setup times.We model the joint admission control/scheduling problem as a Markov decision process (MDP) to gain insight into the optimal control of the production system and use the MDP to benchmark the performance of a simple heuristic acceptance/scheduling policy. Numerical results show that the heuristic performs very well compared with the optimal policy for a wide range of parameter settings, including product family asymmetries in arrival rate, order size, and order reward.     

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.     

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.     

Dynamic scheduling for minimum delay in tandem and parallel constrained queueing models

The optimal scheduling problem in two queueing models arising in multihop radio networks with scheduled link activation is investigated. A tandem radio network is considered. Each node receives exogenous arriving packets which are stored in its unlimited capacity buffer. Links adjacent to the same node cannot transmit simultaneously because of radio interference constraints. The problem of link activation scheduling for minimum delay is studied for two different traffic types. In the first type all packets have a common destination that is one end-node of the tandem. In this case the system is modeled by a tandem queueing network with dependent servers. The server scheduling policy that minimizes the delay is obtained. In the second type of traffic, the destination of each packet is an immediate neighbor of the node at which the packet enters the network. In this case the system corresponds to a set of parallel queues with dependent servers. It is shown that the optimal policy activates the servers so that the maximum number of packets are served at each slot.     

Scheduling policies in the <Emphasis Type="Italic">M</Emphasis>/<Emphasis Type="Italic">G</Emphasis>/1 make-to-stock queue

In this paper, we analyse a production/inventory system modelled as an M/G/1 make-to-stock queue producing different products requiring different and general production times. We study different scheduling policies including the static first-come-first-served, preemptive and non-preemptive priority disciplines. For each static policy, we exploit the distributional Little's law to obtain the steady-state distribution of the number of customers in the system and then find the optimal inventory control policy and the cost. We additionally provide the conditions under which it is optimal to produce a product according to a make-to-order policy. We further extend the application area of a well-known dynamic scheduling heuristic, Myopic(T), for systems with non-exponential service times by permitting preemption. We compare the performance of the preemptive-Myopic(T) heuristic alongside that of the static preemptive-bμ rule against the optimal solution. The numerical study we have conducted demonstrates that the preemptive-Myopic(T) policy is superior between the two and yields costs very close to the optimal.     

Dynamic repositioning for vehicle sharing with setup costs

This study considers a multi-period two-region repositioning problem with setup repositioning costs involved for vehicle sharing systems. We find that incorporating such costs can influence the total cost significantly and complicate the structure of the optimal policy. Moreover, we manage to partially characterize the optimal policy, and then develop an easy-to-implement heuristic policy. The performance of the heuristic policy and the influence of setup repositioning costs on policies are assessed numerically.     

Fluid approximation of a controlled multiclass tandem network

A two-station, four-class queueing network with dynamic scheduling of servers is analyzed. It is shown that the corresponding Markov decision problem converges under fluid scaling to a fluid optimal control model. The structure of the optimal policy for the fluid network, and of an asymptotically optimal policy for the queueing network are derived in an explicit form. They concur with the tandem μ-rule, if this policy gives priority to the same flow of customers in both stations. In general, they are monotone with a linear switching surface. This revised version was published online in June 2006 with corrections to the Cover Date.     

Ergodic control of diffusions with compound Poisson jumps under a general structural hypothesis

We study the ergodic control problem for a class of controlled jump diffusions driven by a compound Poisson process. This extends the results of Arapostathis et al. (2019) to running costs that are not near-monotone. This generality is needed in applications such as optimal scheduling of large-scale parallel server networks.We provide a full characterizations of optimality via the Hamilton–Jacobi–Bellman (HJB) equation, for which we additionally exhibit regularity of solutions under mild hypotheses. In addition, we show that optimal stationary Markov controls are a.s. pathwise optimal. Lastly, we show that one can fix a stable control outside a compact set and obtain near-optimal solutions by solving the HJB on a sufficiently large bounded domain. This is useful for constructing asymptotically optimal scheduling policies for multiclass parallel server networks.