R.D. Nobel

A heuristic rule for routing customers to parallel servers

A practically important problem is the assignment of stochastically arriving service requests to one of several parallel service groups so as to minimize the long-run average sojourn time per service request. An exact solution of this multi-dimensional optimization problem is computationally infeasible. A simple heuristic solution method yielding a good suboptimal rule will be given for the case of server groups with different and generally distributed service times. This solution method is based on a decomposition approach and first principles from Markov decision theory. The main idea of the heuristic method is to apply one step of policy improvement to the best Bernoulli-splitting rule.

Optimal control of a queueing system with heterogeneous servers and setup costs

This paper considers a queueing model with batch Poisson input and two heterogeneous servers, where the service times are exponentially distributed. The faster server is always on, but the slower server is only used when the queue length exceeds a certain level. Activating the slower server involves fixed set-up costs. Also there are linear operating costs and linear holding costs. The class of two-level hysteretic control rules is considered. Rather than proving the overall average cost optimality of a hysteretic rule, the purpose of this paper is to develop a tailor-made policy iteration algorithm for computing the optimal switch-on and switch-off levels for the slower server. An embedding method is used that is generally applicable to structured Markovian control problems with an infinitely large state space.

Practical approximations for finite-buffer queueing models with batch-arrivals

Abstract In this paper we discuss the batch arrival GX/G/1/N queue with a single server and room for only N customers. For this model two different rejection strategies are conceivable: a batch finding upon arrival not enough space in the buffer is rejected completely or the buffer is filled up and only a part of the batch is rejected. For either strategy we are interested in the rejection probabilities both for a batch and for an individual customer. Also we want to investigate the waiting-time distribution for an accepted customer. In general we cannot find analytical solutions for this model. However, by specifying the service-time distribution to be an Erlang-r distribution, a Markov-chain approach is possible and exact results can be obtained. The next step is to get approximate results for the general case via interpolation with respect to the squared coefficient of variation of the service time. We give approximations for the waiting-time percentiles and for the minimal bufferspace such that the rejection probability is below a prespecified level. Also numerical results are given to illustrate the quality of the approximations.

A discrete-time retrial queueing model with one server

This paper presents a one-server queueing model with retrials in discrete-time. The number of primary jobs arriving in a time slot follows a general probability distribution and the different numbers of primary arrivals in consecutive time slots are mutually independent. Each job requires from the server a generally distributed number of slots for its service, and the service times of the different jobs are independent. Jobs arriving in a slot can start their service only at the beginning of the next slot. When upon arrival jobs find the server busy all incoming jobs are sent into orbit. When upon arrival in a slot jobs find the server idle, then one of the incoming jobs (randomly chosen) in that slot starts its service at the beginning of the next slot, whereas the other incoming jobs in that slot, if any, are sent into orbit. During each slot jobs in the orbit try to re-enter the system individually, independent of each other, with a given retrial probability. The ergodicity condition and the generating function of the joint equilibrium distribution of the number of jobs in orbit and the residual service time of the job in service are calculated. From the generating function several performance measures are deduced, like the average orbit size. Also the busy period and the number of jobs served during a busy period are discussed. To conclude, extensive numerical results are presented.

A lost-sales production/inventory model with two discrete production modes

A discrete production/inventory model is considered in which batch orders for a single item arrive at a production facility according to a Poisson process. The items can be produced according to two production modes, regular mode or high speed mode. Changing production mode requires a setup time during which production is disabled. Demand that cannot be satisfied from stock is lost. To control this model with respect to a suitable cost criterion, two-level hysteretic switching strategies are considered. Using generally applicable methods, tractable expressions are obtained for the fraction of lost demand and the average inventory level, amongst others. In these methods an essential role is played by the discrete Fast Fourier Transform

A Mixed Discrete-Time Delay/Retrial Queueing Model for Handover Calls and New Calls Competing for a Target Channel

To study the performance of handover calls approaching a target cell in combination with arrivals of new calls competing for the same cell, a mixed discrete-time delay/retrial model with one server and with priorities for the delayed customers is discussed. The handover calls are modeled as high-priority customers and the new calls as low-priority customers. The priority is non-preemptive. Upon arrival high-priority customers are put in a queue which is served on a first come first served basis. The behavior of the low-priority customers is modeled as in a retrial queue. Arrivals are in batches and all customers are served individually according to generally distributed and independent service times. The joint steady-state distribution of the queue length of the high priority customers and the orbit size of the low-priority customers is studied using probability generating functions. Several performance measures will be calculated, such as the mean queue length of the handover calls and the orbit size of the new calls. Also the covariance between the queue length and the orbit size will be studied, among others.

A discrete-time queueing model with abandonments

This paper presents a multi-server queueing model with abandonments in discrete time. In every time slot a generally distributed number (batch) of customers arrives. The different numbers of arrivals in consecutive slots are mutually independent. Each customer requires a geometrically distributed service time. Customers arriving in a slot can start their service only at the beginning of the next slot. When upon arrival customers find all servers busy, they join a queue and wait for their service. When upon arrival customers find server(s) idle, then as many of the incoming customers (randomly chosen) start their service at the beginning of the next slot as there are idle servers, whereas the excess incoming customers, if any, join the queue. The customers in the queue are served in the order of arrival. Every time slot each customer waiting in the queue decides to abandon the system forever with a fixed (abandon) probability, i.e. independently from the other customers waiting in the queue, his position in the queue or the elapsed waiting time. Arrivals have precedence over departures. The steady-state behaviour of this system is studied.

Optimal Routing of Customers to Parallel Service Groups

This paper deals with a queueing system with two service groups each having its own queue. Customers arrive according to a Poission process and arriving customers must be routed irrevocably to one of the two queues. Using Markov decision theory, we study the question of how to rout newly arriving customers. The paper discusses also the performance of several heuristic routing rules including the shortest expected delay rule.

A Retrial Queueing System with a Variable Number of Active Servers: Dynamic Manpower Planning in a Call Center

A retrial queueing model is considered with Poisson input and an unlimited number of servers. At any epoch only a finite number of the servers are active, the others are called dormant. An active server is always in one of two possible states, idle or busy. When upon arrival of a customer at least one of the active servers is idle, the newly arrived customer goes into service immediately, making the idle server busy. When at an arrival epoch all active servers are busy, the decision must be made to send the newly arrived customer into orbit, or to activate a dormant server for immediate service of the arrived customer. Customers in orbit try to reenter the system after an exponentially distributed retrial time. At service completion epochs the decision must be made to keep the newly become idle server active, or to make this server dormant. The service times of the customers are independent and have a Coxian-2 distribution. Given specific costs for activating servers, keeping servers active and a holding cost for customers staying in orbit, the problem is when to activate and shut down servers in order to minimize the long-run average cost per unit time. Using Markov decision theory an efficient algorithm is discussed for calculating an optimal policy.

The Priority of Inbound Calls over Outbound Calls Modeled as a Discrete-Time Retrial/Delay System.

A one-server discrete-time queueing model is studied with two arrival streams. Both arrival streams are in batches and we distinguish between a stream of low-priority customers, who are put in a queue which is served on a first-come-first-served basis, and a stream of (primary) high-priority customers, who are served uninterruptedly when the batch of high-priority customers finds the server idle upon arrival. High-priority customers are treated as retrial customers, but once in the orbit they lose their high-priority status. The Late Arrival Setup is chosen with Delayed Access. The high-priority retrial customers can be interpreted as inbound calls, and the low-priority customers as outbound calls in a call-center. The joint steady-state distribution of the queue length of the low-priority customers and the orbit size of secondary retrial customers is studied using probability generating functions. Several performance measures will be calculated, such as the mean queue length of the low-priority customers and the orbit size of the secondary retrial customers.