Approximate behavior of tandem queues
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Bibliographic Information
Approximate behavior of tandem queues
（Lecture notes in economics and mathematical systems, 171）
Springer-Verlag, 1979
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Includes bibliographical references
Description and Table of Contents
Description
The following monograph deals with the approximate stochastic behavior of a system consisting of a sequence of servers in series with finite storage between consecutive servers. The methods employ deterministic queueing and diffusion approximations which are valid under conditions in which the storages and the queue lengths are typically large compared with 1. One can disregard the fact that the customer counts must be integer valued and treat the queue as if it were a (stochastic) continuous fluid. In these approximations, it is not necessary to describe the detailed probability distribution of service times; it suffices simply to specify the rate of service and the variance rate (the variance of the number served per unit time). Specifically, customers are considered to originate from an infinite reservoir. They first pass through a server with service rate ~O' vari ance rate ~O' into a storage of finite capacity c . They then pass l through a server with service rate ~l' variance rate ~l' into a storage of capacity c ' etc., until finally, after passing through an nth server, 2 they go into an infinite reservoir (disappear). If any jth storage become , n , the service at the j-lth server is interrupted full j = 1, 2, and, of course, if a jth storage becomes empty the jth server is inter rupted; otherwise, services work at their maximum rate.
Table of Contents
- I. General Theory.- 1. Introduction.- 2. Graphical Representations and Deterministic Approximation.- 3. Motion of Holes.- 4. Diffusion Equation.- 5. Queue Length Distribution.- 6. Soft Boundaries.- 7. Moments.- References.- II. A Single Server.- 1. Diffusion Equation.- 2. Queue Distribution.- 3. Service Rates.- 4. Longtime Behavior of the Joint Distributions.- 5. Service Variances.- 6. Image Solution c1 = ?.- 7. Longtime Behavior c1 = ?.- 8. Discussion.- III. Equilibrium Queue Distributions Two Servers, ?0 = ?1 = ?2, Theory.- 1. Introduction.- 2. Formulation.- 3. Conformal Mappings.- 4. Marginal Distributions.- 5. Symmetry.- 6. Saddle Points and Singularities.- 7. One Large Storage.- 8. Expansions of the Marginal Distributions.- References.- IV. Equilibrium Queue Distributions, Two Servers ?0 = ?1 = ?2, Numerical Results.- 1. Introduction.- 2. Marginal Distributions for c2 = ?.- 3. Relation between c*1, c*2 and w1, w3.- 4. Marginal Distributions c*1 c*2 < ?.- 5. The Service Rate.- 6. Joint Distributions.- V. Time-dependent Solutions ?0 = ?1 = ?2.- 1. Introduction.- 2. Image Solution.- 3. Time-dependent Queue Distribution.- VI. Laplace Transform Methods, Equilibrium Queue Distributions for n = 2, ?0 < ?1 ? ?2.- 1. Analysis of Transforms.- 2. Equilibrium Distributions c1 = c2 = ?, ?0 = ?2 = 0.- 3. Numerical Evaluations.- 4. Equilibrium Distributions c1 = c2 = ?.- 5. Other Special Cases.- 6. Interpretation.- VII. Equilibrium Queue Distributions
- n=2
- ?1 < ?0, ?2
- c1 c2 ??.- 1. Introduction.- 2. Joint Distribution for ?0 = ?2 = 0.- 3. Joint Distribution for ?0, ?2 > 0.- 4. Service Rate for Large But Finite c1, c2.- VIII. Epilogue.- 1. What Was the Question?.- 2. Graphical Representations.- 3. Diffusion Approximations.- 4. A Single Server.- 5. Joint Probability Density for Q1 Q2.- Notation.
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