On Regenerative Processes in Queueing Theory

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.