The Congested Median Problem

The median problem has been generalized to include queueing-like congestion of facilities (which are assumed to have finite numbers of servers). In one statement of the problem, a closest available server is assumed to handle each service request. More general server assignment policies are allowed, however. The analysis requires keeping track of the states (available or unavailable) of all servers. Paralleling the standard deterministic median problem, the objective is to minimize the expected travel time associated with a random service request, weighted appropriately by the equilibrium state probabilities of the system. Under suitable conditions, it is shown that at least one set of optimal locations exists solely on the nodes of the network. This analysis ties together previously disparate efforts in network analysis and spatial queueing analysis.Prepared under Grant Number 78NI-AX-0007 from the National Institute of Law Enforcement and Criminal Justice, Law Enforcement Assistance Administration, U.S. Department of Justice

Scheduling Workforce and Workflow in a Service Factory

We define a service factory to be a network of service-related-workstations, at which assigned workers process work-in-progress that flows through the workstations. Examples of service factory work include mail processing and sorting, check processing and telephoned order processing. Exogenous work may enter the factory at any workstation according to any time-of-day profile. Work-in-progress flows though the factory in discrete time according to Markovian routings. Workers, who in general are cross trained, may work part time or full time shifts, may start work only at designated shift starting times, and may change job assignments at midshift. In order to smooth the flow of work-in-progress through the service factory, work-in-progress may be temporarily inventoried (in buffers) at work stations. The objective is to schedule the workers (and correspondingly, the workflow) in a manner that minimizes labor costs subject to a variety of service-level, contractural and physical constraints. Motivated in part by analysis techniques of discrete time linear time-invariant (LTI) systems, an object-oriented linear programming (OOLP) model is developed. Using exogenous input work profiles typical of large U. S. mail processingfacilities, illustrative computational results are included

An N Server Cutoff Multi-Priority Queue

Consider a multi-priority, nonpreemptive, N-server Poisson arrival queueing system. Service times are negative exponential. In order to save available servers for higher priority customers, arriving customers of each lower priority are deliberately queued whenever the number of servers busy equals or exceeds a given priority-dependent cutoff number. A queued priority i customer enters service the instant there are fewer than the respective cutoff number of servers busy and all higher priority queues are empty. The principal result is the priority i waiting time mean, second moment, and distribution (in transforms). The analysis is extended to systems in which any subset of priority levels may overflow to some other system, rather than join infinite capacity queues. The paper concludes with illustrative computational results

An N Server Cutoff Priority Queue Where Customers Request a Random Number of Servers

Consider a multi-priority, nonpreemptive, N-server Poisson arrival queueing system. The number of servers requested by an arrival has a known probability distribution. Service times are negative exponential. In order to save available servers for higher priority customers, arriving customers of each lower priority are deliberately queued whenever the number of servers busy equals or exceeds a given priority-dependent cutoff number. A queued priority i customer enters service the instant the number of servers busy is at most the respective cutoff number of servers minus the number of servers requested (by the customer) and all higher priority queues are empty. In other words the queueing discipline is in a sense HOL by priorities, FCFS within a priority. All servers requested by a customer start service simultaneously; service completion instants are independent. We derive the priority i waiting time distribution (in transform domain) and other system statistics

Efficient Computation of Probabilities of Events Described by Order Statistics and Applications to Queue Inference

This paper derives recursive algorithms for efficiently computing event probabilities related to order statistics and applies the results in a queue inferencing setting. Consider a set of N i.i.d. random variables in [0, 1]. When the experimental values of the random variables are arranged in ascending order from smallest to largest, one has the order statistics of the set of random variables. Both a forward and a backward recursive O(N3 ) algorithm are developed for computing the probability that the order statistics vector lies in a given N-rectangle. The new algorithms have applicability in inferring the statistical behavior of Poisson arrival queues, given only the start and stop times of service of all N customers served in a period of continuous congestion. The queue inference results extend the theory of the "Queue Inference Engine" (QIE), originally developed by Larson in 1990 [8]. The methodology is extended to a third O(N 3 ) algorithm, employing both forward and backward recursion, that computes the conditional probability that a random customer of the N served waited in queue less than r minutes, given the observed customer departure times and assuming first come, first served service. To our knowledge, this result is the first O(N3 ) exact algorithm for computing points on the in-queue waiting time distribution function,conditioned on the start and stop time data. The paper concludes with an extension to the computation of certain correlations of in-queue waiting times. Illustrative computational results are included throughout

Efficient Computation of Probabilities of Events Described by Order Statistics and Application to a Problem of Queues

Consider a set of N i.i.d. random variables in [0, 1]. When the experimental values of the random variables are arranged in ascending order from smallest to largest, one has the order statistics of the set of random variables. In this note an O(N3) algorithm is developed for computing the probability that the order statistics vector lies in a given rectangle. The new algorithm is then applied to a problem of statistical inference in queues. Illustrative computational results are included

Using Partial Queue-Length Information to Improve the Queue Inference Engine's Performance

The Queue Inference Engine (QIE) uses queue departure time data over a single congestion period to infer queue statistics. With partial queue-length information, the queue statistics become more accurate and the computational burden is reduced. We first consider the case in which we are given that the queue length never exceeded a given length L. We then consider the more general case in which we are given the times of all L-to-(L + 1) and (L + 1)-to-L queue-length transitions. We present algorithms, parallel to the QIE algorithms,for deriving the queue statistics under the new conditioning information. We also present computational results, comparing both accuracy and computation time, under the QIE and the new algorithms, for several sample runs

Deducing Queue Statistics from Transactional Data

Revised May 1988The transactional data of a queueing system are the recorded times of service commencement and service completion for each customer served. With increasing use of computers to aid or even perform service one often has machine readable transactional data, but virtually no information about the queue itself. In this paper we propose a way to deduce the queueing behavior of Poisson arrival queueing systems from only the transactional data and the Poisson assumption. For each congestion period in which queues may form, the key quantities obtained are mean wait in queue, time-dependent mean number in queue, and probability distribution of the number in queue observed by a randomly arriving customer. The methodology builds on arguments of order statistics and usually requires a computer to evaluate a recursive function. The paper concludes with a proposed procedure for estimating the extent of balking and/or reneging present in a queueing system