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Gated Infinite Server Queues in Light Traffic

Dimitra Pinotsi, Michael A. Zazanis

Abstract

We consider $M/G/\infty$ queues with gated service and obtain results on the distribution of the stage length and the number of customers served in a stage when the system is stationary. The stage length density is expressed as an infinite series of terms, involving the solution of an infinite system of linear equations. The convergence of a sequence of solutions arising from truncations of the infinite system is established in the light traffic case. Analogous results are established for a similar $GI/M/\infty$ gated system.

Gated Infinite Server Queues in Light Traffic

Abstract

We consider queues with gated service and obtain results on the distribution of the stage length and the number of customers served in a stage when the system is stationary. The stage length density is expressed as an infinite series of terms, involving the solution of an infinite system of linear equations. The convergence of a sequence of solutions arising from truncations of the infinite system is established in the light traffic case. Analogous results are established for a similar gated system.
Paper Structure (10 sections, 10 theorems, 96 equations, 4 figures)

This paper contains 10 sections, 10 theorems, 96 equations, 4 figures.

Table of Contents

  1. Introduction: The gated, exhaustive, parallel service system
  2. The gated $M/GI/\infty$ system and the stage length density
  3. Exponential service times
  4. The infinite linear system (\ref{['y_system']})
  5. The synchronized gated $GI/M/\infty$ system - number of customers served in a stage
  6. The light traffic case
  7. The infinite linear system (\ref{['Poisson_system']})
  8. Appendix
  9. Infinite linear systems and strictly diagonally dominant matrices
  10. Markov chains on general state spaces

Key Result

Proposition 1

The sequence of consecutive stage lengths $\{Y_n\}$ is a Markov chain with state space ${\sf X}:=[0,\infty)$ and transition kernel $Q(x,A)= \int_A q(x,y)dy$ for $x \in {\sf X}$, $A \in \mathcal{B}$, which is absolutely continuous with respect to the Lebesgue measure for all $x$ with density

Figures (4)

  • Figure 1: A series representation for the stationary density of the length of a stage in a gated $M/M/\infty$ queue in light traffic. Here $\lambda=1$ and $\mu =2.5$. A truncated ($10 \times 10$) version of the system (\ref{['is']}) is used to obtain approximate values for $\beta_2,\ldots,\beta_{11}$. These in turn are used in (\ref{['f-moments']}) to give a numerical illustration of the rapidity of convergence. A comparison with the actual density of the stage length in shown in Figure \ref{['fig3']}.
  • Figure 2: A comparison of the actual density for the invariant density of the stage length on an $M/M/\infty$ queue in light traffic, as obtained by simulation with the solution given by (\ref{['invariant-density-m']}). Notice the deterioration and, eventually, the invalidity of the quality of the light traffic approximation when $\rho$ becomes larger.
  • Figure 3: The mean stage length in the Gated $M/M/\infty$. The dotted line is obtained by simulating $10^4$ stages. The red line, gives the light traffic approximation (equation \ref{['Exp-MeanStageLength']}). As the traffic intensity $\rho$ increases, the quality of the light traffic approach deteriorates.
  • Figure 4: Arrivals are Poisson ($\lambda$) and $\rho:=\lambda/\mu=0.85$. The system (\ref{['xm']}) is truncated at $N=100$ and similarly 100 terms are taken in the series (\ref{['pi-final']}).

Theorems & Definitions (19)

  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • Proposition 3
  • Proposition 4
  • proof
  • Remark 5
  • Theorem 6: Foster--Lyapunov Criterion
  • Proposition 7
  • ...and 9 more