Elements of Queueing Theory: Palm Martingale Calculus and by François Baccelli, Pierre Brémaud (auth.)

By François Baccelli, Pierre Brémaud (auth.)

The Palm concept and the Loynes concept of desk bound structures are the 2 pillars of the fashionable method of queuing. This publication, offering the mathematical foundations of the idea of desk bound queuing structures, includes a thorough remedy of either one of these.

This procedure is helping to explain the image, in that it separates the duty of acquiring the main approach formulation from that of proving convergence to a desk bound kingdom and computing its law.

The conception is continually illustrated by way of classical effects and types: Pollaczek-Khintchin and Tacacs formulation, Jackson and Gordon-Newell networks, multiserver queues, blockading queues, loss structures etc., however it additionally comprises contemporary and critical examples, the place the instruments built turn into indispensable.

Several different mathematical instruments that are worthwhile inside of this method also are awarded, akin to the martingale calculus for element techniques, or stochastic ordering for desk bound recurrences.

This completely revised moment variation includes vast additions - particularly, routines and their strategies - rendering this now vintage reference compatible to be used as a textbook.

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Additional info for Elements of Queueing Theory: Palm Martingale Calculus and Stochastic Recurrences, 2nd Edition

Example text

AER, [T1] = 1. 14) P( -To> w) =A ioo 25 (1- Fo(u))du. 15) P(T1 > v) =A 1= (1- F0 (u))du. Thus -To and T1 are identically distributed under P. f. f. F0 . 5 The Slivnyak Inverse Construction Let {(h}, t E IR, be a flow on ( J2, F), and N be a point process which is compatible with {Bt}. 16) where J2o ={To= 0}. 18) (i) 0 < E 0 [T1] < oo, (ii) P 0 [T1 >OJ= 1, (iii) E 0 [N(O, t]] < oo, \ft < t 0 , for some to > 0. We shall see that P 0 is then the Palm probability PRr associated with the stationary point process (N, Bt, P), for some probability P which is Bt-invariant for all t E R Moreover, in view of the inversion formula, P will be unique.

The Palm Calculus of Point Processes (for instance N counts all the discontinuity points of {Y(t)} and Y'(t) is the derivative of Y(t) between discontinuity points). AER,[Y(O)- Y(O- )] = 0. The above equality constitutes the (rate) conservation principle. We shall see many applications of this elementary formula in Chapter 3. 4. Observe that the boundedness condition for {Y(t)} can be replaced by one of the following three conditions: (a) E[l Y(O) I] < oo and E[l Y'(O) IJ < oo; (b) E[l Y(O) I] < oo and ER,[I Y(O)- Y(O-) IJ < oo; (c) E[l Y'(O) IJ < oo and ER,[I Y(O)- Y(O-) IJ < oo.

3 Basic Formulas of Palm Calculus 27 The interchange is justified by Lebesgue's theorem in view of the bound ! 18,iii). < oo. So, we can define the Palm probability Pt associated with (N, Bt, P). Let us conclude by proving that P 0 =Pt. 22) since on D 0 , Bra o Bt is the identity for all t E (0, T1]. e. 19): Therefore E 0 [T11A] = ER_,[T11A], A E F. Since by construction, T1 > 0, P 0 =Pt. 3. d. 18,iii). Td o Bv], where V is a random variable which, 'conditionally upon everything else', is uniformly distributed on [0, T1] (for the above to make sense, we must of course enlarge the probability space).

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