Approximate Behavior of Tandem Queues

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. ASingle Server.- 5. Joint Probability Density for Q1 Q2.- Notation.