A compensation approach for queueing problems

No abstract

A two node Jackson network with infinite supply of work

We consider a Jackson network with two nodes, with no exogenous input, but instead an infinite supply of work at each of the nodes: whenever a node is empty, it processes a job from this infinite supply. We obtain an explicit expression for the steady state distribution of this system, as an infinite sum of product forms

Shortest Expected Delay Routing for Erlang Servers

The queueing problem with a Poisson arrival stream and two identical Erlang servers is analysed for the queueing discipline based on shortest expected delay. This queueing problem may be represented as a random walk on the integer grid in the first quadrant of the plane. In the paper it is shown that the equilibrium distribution of this random walk can be written as a countable linear combination of product forms. This linear combination is constructed in a compensation procedure. In this case the compensation procedure is essentially more complicated than in other cases where the same idea was exploited. The reason for the complications is that in this case the boundary consists of several layers which in turn is caused by the fact that transitions starting in inner states are not restricted to end in neighbouring states. Good starting solutions for the compensation procedure are found by solving the shortest expected delay problem with the same service distributions but with instantaneous jockeying. It is also shown that the results can be used for an efficient computation of relevant performance criteria

Analyzing GI/E_r/1 queues

In this paper we study a single-server system with Erlang-r distributed service times and arbitrarily distributed interarrival times. It is shown that the waiting-time distribution can be expressed as a finite sum of exponentials, the exponents of which are the roots of an equation. Under certain conditions for the interarrival-time distribution, this equation can be transformed to r contraction equations, the roots of which can be easily found by successive substitutions. The conditions are satisfied for several practically relevant arrival processes. The resulting numerical procedures are easy to implement and efficient and appear to be remarkably stable, even for extreme high values of r and for values of the traffic load close to 1. Numerical results are presented

Mean value approximation for closed queueing networks with multi server stations

We fonnulate an approximate recursive relation for the sojourn time, queuelength and throughput of a multi server station, embedded in a closed queueing network. Based on that relation, we derive a mean value approximation and formulate a Schweitzer approximation for solving large networks. Numerical examples show that the approximate mean value algorithm yields accurate results

Fifteen years of experience with modelling courses in the Eindhoven program of applied mathematics

The curriculum of Applied Mathematics at the Eindhoven University of Technology (TU/e) in the Netherlands includes a series of modelling projects: the so-called Modelling Track. This track was introduced in the curriculum fifteen years ago and has a specific educational approach. Mathematics that may be useful in the projects is not necessarily taught in courses preceding the modelling projects. Moreover, during the projects, students have to use their current skills and knowledge or even have to learn (or discover) new techniques by themselves. Overall, this teaching method has been quite successful in terms of students¿ results and satisfaction. Project coaches have always been recruited from the entire department and gradually, the majority of the staff has become pleased with this method of modeling education. Throughout fifteen years, the structure and content of the series of projects have evolved. This article is based on reflections concerning these changes expressed by the coordinator of the mathematical modelling education (the second author) and the educational advisor (the first author), both of whom have been closely involved in the track¿s development over the years. These changes will be described and their external as well as internal causes will be identified. Examples of external causes are developments of technical phenomena in society, university-wide educational innovations, and a change in the overall structure of the university¿s academic calendar. An example of an internal cause is the variety in background of the project coaches. Finally, strengths and weaknesses of the track will be analyzed. The purpose of the article is to share the experiences with this way of teaching mathematical modelling in higher education and give advice to others who want to implement it

Exact FCFS matching rates for two infinite multi-type sequences

We consider an infinite sequence of items of types C = {c_1, ..., c_I}, and another infinite sequence of items of types S = {s_1, ... , s_J}, and a bipartite graph G of allowable matches between the types. Matching the two sequences on a first come first served basis defines a unique infinite matching between the sequences. For (c_i, s_j) in G we define the matching rate r_(c_i,s_j) as the long term fraction of (c_i, s_j) matches in the infinite matching, if it exists. We assume that the types of items in the two sequences are i.i.d. with given probability vectors a, ß. We describe this system by a Markov chain, obtain conditions for ergodicity, and derive its stationary distribution which is of product form. We show that if the chain is ergodic, then the matching rates exist almost surely, and give a closed form formula to calculate them

The M/M/c with critical jobs

We consider the M/M/c queue, where customers transfer to a critical state when their queueing (sojourn) time exceeds a random time. Lower and upper bounds for the distribution of the number of critical jobs are derived from two modifications of the original system. The two modified systems can be efficiently solved. Numerical calculations indicate the power of the approach