On a two-server finite queuing system with ordered entry and deterministic arrivals

Consider a two-server, ordered entry, queuing system with heterogeneous servers and finite waiting rooms in front of the servers. Service times are negative exponentially distributed. The arrival process is deterministic. A matrix solution for the steady state probabilities of the number of customers in the system is derived. The overflow probability will be used to formulate the stability condition of a closed-loop conveyor system with two work stations.     

Developments in operational research: R.W. Eglese and G.K. Rand (eds.) Pergamon, Oxford, 1984, vii + 110 pages, $20.00

A practical method of calculating the distribution of the number of customers in the single server queueing system with inhomogeneous arrival rate and discrete service time distribution is proposed. The system is formulated as an inhomogeneous Markov chain in discrete time, leading to recurrence relations for the state probabilities. The recurrence relations are then solved numerically. Various measures of performance, such as mean and variance of the number of customers in the system and virtual waiting time can be obtained from these results. Examples are presented to demonstrate the scope of the method, including time-dependent behaviour of homogeneous queues; cyclic behaviour of queues with cyclic arrival rates; and a previously published study of an airport runway in which the author had to resort to crude interpolation to obtain results. The method can be further extended to provide a reasonably accurate approximation for some systems with continuous distributions of service times.     

Analysis of the M/Gi/1 →/M/1 queueing model

An M/GI/1 queueing system is in series with a unit with negative exponential service times and infinite waiting room capacity. We determine a closed form expression for the generating function of the joint queue length distribution in steady state. This result is obtained via the solution of a new type of functional equation in two variables.     

Augmented truncation approximations of discrete-time Markov chains

Let P be a positive recurrent infinite transition matrix with invariant distribution π and be a truncated and arbitrarily augmented stochastic matrix with invariant distribution _(n)π. We investigate the convergence ‖_(n)π−π‖→0, as n→∞, and derive a widely applicable sufficient criterion. Moreover, computable bounds on the error ‖_(n)π−π‖ are obtained for polynomially and geometrically ergodic chains. The bounds become rather explicit when the chains are stochastically monotone.     

Optimal incentive-compatible pricing for M/G/1 queues

This paper extends the applicability of the pricing results of Mendelson and Whang (Oper. Res. 38 (5) (1990) 870) and Balachandran and Radhakrishnan (Management Sci. 40 (10) (1994) 353) for congested service facilities by considering general, class-dependent, service time distributions. Two theorems for nonpreemptive M/G/1 queues and preemptive-resume M/G/1 queues are presented.     

Mean sojourn times for phase-type discriminatory processor sharing systems

In a discriminatory processor sharing (DPS) queueing model, each job (or customer) belongs to one out of finitely many classes. The arrival processes are Poisson. Classes differ with respect to arrival rates and service time distributions. Moreover, classes have different priority levels. All jobs present are served simultaneously but the fraction of the server’s capacity allocated to each one of them is proportional to their class priority parameter (while the total capacity is of course fixed).     

Analysis of job transfer policies in systems with unreliable servers

We consider a system where incoming jobs may be executed at different servers, each of which goes through alternating periods of being available and unavailable. Neither the states of the servers nor the relevant queue sizes are known at moments of arrival. Hence, a load balancing mechanism that relies on random time-out intervals and job transfers from one queue to another is adopted. The object is to minimize a cost function which may include holding costs and transfer costs. A model of a single queue with an unreliable server and timeouts is analyzed first. The results are then used to obtain an approximate solution for arbitrary number of queues. Several transfer policies are evaluated and compared.     

Simulating hospital emergency departments queuing systems: (GI/G/m(t)) : (IHFF/N/∞)

The queuing system to be considered in this paper is a real case study with the following characteristics: (1) general independent interarrival distribution, (2) general service-time distribution, (3) limited waiting room, (4) patient's priorities increase up to a certain number, (5) time-dependent number of servers (doctors), (6) infinite patient population, (7) each server meets the system only once within a certain period of time, while the total number of the available servers is known. The system to be considered is a hospital's Emergency Department and a general simulation algorithm is presented as well as the system's operating characteristics. This algorithm, implemented on a mini-computer or an inexpensive microcomputer solved a sophisticated operations research problem.     

Impedance-effects in capacity estimation

Formulas for the capacity of an unsignalized intersection for traffic streams which are subordinated to two or more streams of higher priority are developed. The case of only two conflicting streams analysed by Tanner (1962) is extended. The process at the intersection is modelled as a priority queuing system with Poisson input and service times depending on the work load.