Optimal threshold policies in a two-class preemptive priority queue with admission and termination control

We consider a two-class 1 preemptive priority queue in which there are two essential, on-line decisions that have to be taken. The first is the decision to either accept or reject new type-1 or type-2 jobs. The second is the decision to abort jobs, i.e., to remove any type-1 or type-2 jobs from the system. We show that there exist optimal threshold policies for these two types of, decisions.     

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.     

A single-server batch arrival queue with returning customers

We consider a new class of batch arrival retrial queues. By contrast to standard batch arrival retrial queues we assume if a batch of primary customers arrives into the system and the server is free then one of the customers starts to be served and the others join the queue and then are served according to some discipline. With the help of Lyapunov functions we have obtained a necessary and sufficient condition for ergodicity of embedded Markov chain and the joint distribution of the number of customers in the queue and the number of customers in the orbit in steady state. We also have suggested an approximate method of analysis based on the corresponding model with losses.     

Discrete-time queue with Bernoulli bursty source arrival and generally distributed service times

In this contribution, a discrete-time single-server infinite-capacity queue with correlated arrivals and general service times is investigated. Arrivals of cells are modelled as an on/off source process with geometrically distributed on-periods and off-periods, which is called Bernoulli bursty source. Based on the probability generating function technique, closed-form expression of some performance measures of system, such as average buffer content, unfinished work, cell delay and so on, are obtained. Finally, the effects of system parameters on performance measures are illustrated by some numerical examples.     

A queueing model for general group screening policies and dynamic item arrivals

Classification of items as good or bad can often be achieved more economically by examining the items in groups rather than individually. If the result of a group test is good, all items within it can be classified as good, whereas one or more items are bad in the opposite case. Whether it is necessary to identify the bad items or not, and if so, how, is described by the screening policy. In the course of time, a spectrum of group screening models has been studied, each including some policy. However, the majority ignores that items may arrive at random time epochs at the testing center in real life situations. This dynamic aspect leads to two decision variables: the minimum and maximum group size. In this paper, we analyze a discrete-time batch-service queueing model with a general dependency between the service time of a batch and the number of items within it. We deduce several important quantities, by which the decision variables can be optimized. In addition, we highlight that every possible screening policy can, in principle, be studied, by defining the dependency between the service time of a batch and the number of items within it appropriately.     

Optimal regulation of the service rate for a queue with finite waiting space

The optimal control for the service rate of a single-server queue with limited waiting space is derived. Costs are associated with providing the service and with waiting. A comparison with the traditional steady-state result is made.     

Analysis of the adaptive MMAP[K]/PH[K]/1 queue: A multi-type queue with adaptive arrivals and general impatience

In this paper we introduce the adaptive MMAP[K] arrival process and analyze the adaptive MMAP[K]/PH[K]/1 queue. In such a queueing system, customers of K different types with Markovian inter-arrival times and possibly correlated customer types, are fed to a single server queue that makes use of r thresholds. Service times are phase-type and depend on the type of customer in service. Type k customers are accepted with some probability a_i,k if the current workload is between threshold i − 1 and i. The manner in which the arrival process changes its state after generating a type k customer also depends on whether the customer is accepted or rejected.     

Optimal control of a queue under a quality-of-service constraint with bounded and unbounded rates

We consider a simple Markovian queue with Poisson arrivals and exponential service times for jobs. The controller chooses state-dependent service rates from an action space. The queue has a finite buffer, and when full, new jobs get rejected. The controller’s objective is to choose optimal service rates that meet a quality-of-service constraint. We solve this problem analytically and compute it numerically under two cases: When the action space is unbounded and when it is bounded.     

Maximizing throughput in finite-source parallel queue systems

Motivated by the dispatching of trucks to shovels in surface mines, we study optimal routing in a Markovian finite-source, multi-server queueing system with heterogeneous servers, each with a separate queue. We formulate the problem of routing customers to servers to maximize the system throughput as a Markov Decision Process. When the servers are homogeneous, we demonstrate that the Shortest Queue policy is optimal, and when the servers are heterogeneous, we partially characterize the optimal policy and present a near-optimal and simple-to-implement policy. We use the model to illustrate the substantial benefits of pooling, by comparing it to the permanent assignment of customers to servers.     

Bias optimality of admission control in a non-stationary repairable queue

We study a non-stationary repairable queue with a single server and multiple customers’ types. The difference between types of customers is defined by the offered rewards. We show that the bias optimal policy has the trunk reservation (threshold) form. Furthermore, under some given conditions, we also prove that the control level of the bias optimal policy is monotone about time.     

Optimal admission control in queues with abandonments

We consider a multi-sever Markovian queueing system with abandonments where admitted customers pay a reward either at the time of arrival or service completion. There is a cost associated with abandonments and a holding cost associated with customers in the system. We prove that the policy that maximizes the long-run average reward is of threshold type and completely characterize the optimal thresholds. We conclude with a comparison of various characteristics of the two variants of the model.     

Optimal control policies for ambulance diversion

Ambulance diversion (AD) is used by emergency departments (EDs) to relieve congestion by requesting ambulances to bypass the ED and transport patients to another facility. We study optimal AD control policies using a Markov Decision Process (MDP) formulation that minimizes the average time that patients wait beyond their recommended safety time threshold. The model assumes that patients can be treated in one of two treatment areas and that the distribution of the time to start treatment at the neighboring facility is known. Assuming Poisson arrivals and exponential times for the length of stay in the ED, we show that the optimal AD policy follows a threshold structure, and explore the behavior of optimal policies under different scenarios. We analyze the value of information on the time to start treatment in the neighboring hospital, and show that optimal policies depend strongly on the congestion experienced by the other facility. Simulation is used to compare the performance of the proposed MDP model to that of simple heuristics under more realistic assumptions. Results indicate that the MDP model performs significantly better than the tested heuristics under most cases. Finally, we discuss practical issues related to the implementation of the policies prescribed by the MDP.     

Optimal threshold policies in a workload model with a variable number of service phases per job

We consider an M|E _N |1 queue in which there are two essential on-line decisions that have to be taken. The first one is the decision to either accept or reject new jobs. The second one is the decision to either continue or abort the service of a job. We show that under certain regularity conditions, there exist optimal threshold policies for these two types of decisions.     

Analysis of optimal and nearly optimal sequencing policies for a closed queueing network

We analyze sequencing policies designed to most effectively utilize the resources of a closed queueing network representation of a manufacturing system. A continuous time Markov decision process formulation is used to compare the performance of optimal sequencing policies and a heuristic developed by analyzing a heavy traffic approximation of the system.     

On the optimal control of arrivals to a single queue with arbitrary feedback delay

We consider a problem of admission control to a single queue in discrete time. The controller has access to k step old queue lengths only, where k can be arbitrary. The problem is motivated, in particular, by recent advances in high-speed networking where information delays have become prominent. We formulate the problem in the framework of Completely Observable Controlled Markov Chains, in terms of a multi-dimensional state variable. Exploiting the structure of the problem, we show that under appropriate conditions, the multi-dimensional Dynamic Programming Equation (DPE) can be reduced to a unidimensional one. We then provide simple computable upper and lower bounds to the optimal value function corresponding to the reduced unidimensional DPE. These upper and lower bounds, along with a certain relationship among the parameters of the problem, enable us to deduce partially the structural features of the optimal policy. Our approach enables us to recover simply, in part, the recent results of Altman and Stidham, who have shown that a multiple-threshold-type policy is optimal for this problem. Further, under the same relationship among the parameters of the problem, we provide easily computable upper bounds to the multiple thresholds and show the existence of simple relationships among these upper bounds. These relationships allow us to gain very useful insights into the nature of the optimal policy. In particular, the insights obtained are of great importance for the problem of actually computing an optimal policy because they reduce the search space enormously. This revised version was published online in June 2006 with corrections to the Cover Date.     

Time-dependent analysis of a queue with batch arrivals andn levels of nonpreemptive priority

A queueing system with batch arrivals andn classes of customers with nonpreemptive priorities between them is considered. Each batch arrives according to the Poisson distribution and contains customers of all classes while the service times follow arbitrary distributions with different probability density functions for each class. For such a model the system states probabilities both in the transient and in the steady state are analysed and also expressions for the Laplace transforms of the busy period densities for each class and for the general busy period are obtained.     

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.     

A retrial queue with structured batch arrivals,priorities and server vacations

A retrial queue accepting two types of customers with correlated batch arrivals and preemptive resume priorities is studied. The service times are arbitrarily distributed with a different distribution for each type of customer and the server takes a single vacation each time he becomes free. For such a model the state probabilities are obtained both in a transient and in a steady state. Finally, the virtual waiting time of an arbitrary ordinary customer in a steady state is analysed.     

Optimal patient and personnel scheduling policies for care-at-home service facilities

In this paper we study the problem of personnel planning in care-at-home facilities. We model the system as a Markov decision process, which leads to a high-dimensional control problem. We study monotonicity properties of the system and derive structural results for the optimal policy. Based on these insights, we propose a trunk reservation heuristic to control the system. We provide numerical evidence that the heuristic yields close to optimal performance, and scales well for large problem instances.