Abstract

A c-congestion period of an m/m/∞-queue is a period during which the number of customers in the system is continuously above level c. Interesting quantities related to a c-congestion period are, besides its duration D_c, the total area A_c above c, and the number of arrived customers N_c. In the literature Laplace transforms for these quantities have been derived, as well as explicit formulae for their means. Explicit expressions for higher moments and covariances (between D_c,N_c and A_c), however, have not been found so far.

This paper presents recursive relations through which all moments and covariances can be obtained. Up to a starting condition, we explicitly solve these equations; for instance, we write explicitly in terms of . We then find formulae for these starting conditions (which directly relate to the busy period in the m/m/∞ queue).

Finally, a c-intercongestion period is defined as the period during which the number of customers is continuously below level c. Also for this situation a recursive scheme allows us to explicitly compute higher moments and covariances. Additionally we present the Laplace transform of a so-called intercongestion triple of the three performance quantities. It is also shown that expressions for the quantities of a c-intercongestion period can be used in an approximation for the c-congestion period. This is especially useful as the expressions for the c-intercongestion period are numerically more stable than those for the c-congestion period.

Keywords: m/m/∞; Transient behavior; c-congestion period; Busy period

Article Outline

1. Introduction

1.1. Literature

1.2. Contribution

1.3. Outline

2. Preliminaries

2.1. Definitions

2.2. Decomposition of a passage time into congestion periods

2.3. Analysis of a c-congestion period

3. Quantities of a c-congestion period

3.1. Duration of a c-congestion period

3.1.1. Mean duration of a c-congestion period

3.1.2. Second moment of duration of a c-congestion period

3.1.3. Higher moments of the duration of a c-congestion period

3.2. Number of arriving customers during a c-congestion period

3.2.1. Mean number of arriving customers in a c-congestion period

3.2.2. Second moment of the number of arriving customers

3.3. Area swept above c during a c-congestion period

3.3.1. Mean area swept above c

3.3.2. Second moment of the area swept above c

4. Joint expectations of the C-congestion period quantities

4.1. Joint expectation of the duration and number of arrivals

4.2. Joint expectation of the duration and the area swept above c

4.3. Joint expectation of the number of arrivals and the area swept above c

5. Moments and joint expectations of the busy-period quantities

5.1. Preater’s lt of the 0-congestion triple (D₀,N₀,A₀)

5.2. Moments of the duration of the busy period

5.2.1. First moment

5.2.2. Second moment

5.2.3. Relation between (24) and the results of Liu and Shi [6]

5.3. Joint expectation of the busy period

5.4. Moments for service times other than 1

6. C-intercongestion periods

6.1. Definitions

6.2. Laplace transforms of the duration and the intercongestion triple

6.2.1. Laplace transform of the intercongestion period duration

6.2.2. Laplace transform of c-intercongestion triple

6.3. Moments of the c-intercongestion period quantities

6.3.1. Moments of the duration of an c-intercongestion period

6.3.2. Moments of the number of arrivals during a c-intercongestion period

6.3.3. Moments of the area swept under c during a c-intercongestion period

6.3.4. Joint expectations of a c-intercongestion period

7. Intercongestion period as an approximation of a congestion period

8. Concluding remarks

Acknowledgements

References

Vitae