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# On Regenerative Processes in Queueing Theory Jacob Willem Cohen

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## Summary

The single server queue GIIG/1 1 1. 1 Definitions 1 1. 2 Regenerative processes 2 1. 1 On the stationary distribution of the actual and virtua131 waiting time for the M/G/K queueing system 2. 3 The distribution of the supremum of the virtual waiting 75 * (00) d' b 1 tlme ~t urlng a usy cyc e i.

## On Regenerative Processes in Queueing Theory Summary

### On Regenerative Processes in Queueing Theory by Jacob Willem Cohen

I. The single server queue GIIG/1 1 1. 1 Definitions 1 1. 2 Regenerative processes 2 1. 3 The sequence n 1,2, . . . 4 = !::!n' 1. 4 The process t dO,co)} 11 {~t' The process t dO,co)} 1. 5 15 {~t' Applications to the GIIG/1 queue 1. 6 16 The average virtual waiting time during a busy 17 cycle ii. Little's formula 17 iii. The relation between the stationary distributions 18 of the virtual and actual waiting time iv. The relation between the distribution of the idle 20 period and the stationary distribution of the actual waiting time v. The limiting distribution of the residual service 24 time GBP. , -pw vi. The relation for ~ rn E{e -n} 25 n=O 1. 7 Some notes on chapter I 27 II. The M/G/K system 31 2. 1 On the stationary distribution of the actual and virtua131 waiting time for the M/G/K queueing system 2. 2 The M/G/K loss system 36 2. 3 Proof of Erlang's formula for the M/G/K loss system 43 i. Proof for the system MIMI'" 45 ii. Proof for the system M/G/co 47 VI iii. Proof fol' the MIG IK los s system III. The M/G/1 system 3. 1 Introduction 71 (K) 3. 2 Downcrossings of the ~t -process 74 3. 3 The distribution of the supremum of the virtual waiting 75 * (00) d' b 1 tlme ~t urlng a usy cyc e i. The exit probability 76 ii.

I. The single server queue GI/G/1.- 1.1 Definitions.- 1.2 Regenerative processes.- 1.3 The sequence wn, n = 1,2,....- 1.4 The process {v, t ?[0, ?)}.- 1.5 The process {?t, t ?[0, ?)}.- 1.6 Applications to the GI/G/1 queue.- i. The average virtual waiting time during a busy cycle.- ii. Little's formula.- iii. The relation between the stationary distributions of the virtual and actual waiting time.- iv. The relation between the distribution of the idle period and the stationary distribution of the actual waiting time.- v. The limiting distribution of the residual service time ??.- vi. The relation for $$\sum\limits_{n = 0}^\infty {{r^n}\,E\{ e{\,^{^{ - \rho w}}}^{ - n}} \}$$.- 1.7 Some notes on chapter I.- II. The M/G/K system.- 2.1 On the stationary distribution of the actual and virtual waiting time for the M/G/K queueing system.- 2.2 The M/G/K loss system.- 2.3 Proof of Erlang's formula for the M/G/K loss system.- i. Proof for the system M/M/?.- ii. Proof for the system M/G/?.- iii. Proof for the M/G/K loss system.- III. The M/G/1 system.- 3.1 Introduction.- 3.2 Downcrossings of the vt(K) -process.- 3.3 The distribution of the supremum of the virtual waiting time vt(?) during a busy cycle.- i. The exit probability.- ii. The distribution of the supremum.- 3.4 The distribution of the downcrossings 8l.- 3.5 Derivation of the stationary distribution of the vt(K) - process, I.- 3.6 Derivation of the stationary distribution of the vt(K) - process, II.- 3.7 Some remarks on the actual and virtual waiting time processes.- i. Quasi-stationary distributions.- ii. The sequence wnk, n = 1,2,..., for fixed k.- References.

NLS9783540076278
9783540076278
3540076271
On Regenerative Processes in Queueing Theory by Jacob Willem Cohen
New
Paperback
Springer-Verlag Berlin and Heidelberg GmbH & Co. KG
1976-03-01
93
N/A
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