Download Elements of Queuing Theory: Palm-Martingale Calculus and by Francois Baccelli, Pierre Bremaud PDF

By Francois Baccelli, Pierre Bremaud

This publication comprises an advent to basic probabilistic equipment acceptable to queueing platforms with ergodic inputs. the most features handled are balance (construction of desk bound states and convergence theory), time and purchaser averages (system equations), comparability of provider disciplines and precedence ideas. those contain, between different issues, Loynes' idea and its extensions, the elemental fomulas (L=*LAMBDA*W, L=*LAMBDA*G, Kleinrock's invariance relations), PASTA, insensitivity, and optimality of SPRT. The originality of the presentation lies within the systematic use of recurrence equations to explain the dynamics of the platforms, and of the Palm chance framework to explain the desk bound behaviour of such structures. additionally, the perspective followed is only probabilistic, emphasizing using coupling and pattern direction arguments (ergodic and sub-additive ergodic thought, trajectory recognition of stochastic orders). The ebook includes an creation at the idea of palm likelihood at the actual line, and indicates some of the connections with the speculation of stochastic depth.

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Extra info for Elements of Queuing Theory: Palm-Martingale Calculus and Stochastic Recurrences (Applications of Mathematics)

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T h e input process is defined on the Palm space of a stationary point process a s for G / G / l / m queues. There are s 2 1 servers attending customers and the allocation rule is that an arriving customer is assigned to the server with the smallest workload. Once assigned, this customer will wait and then be served a t unit rate until completion. We will construct the stationary workload process {W,),n E Z, where . , W i ) is a permutation in increasing order of the workload found in each server by the n t h customer upon arrival : WA W: ...

1). 1) is that, on &, the (P,Ft)-intensity and the ( P k , Ft)-intensity coincide, that is, for all (a, b] c & , A E &, Proof: By definition of Palm probability, the left-hand side of the above equal- ity is < a t is the original point processes Nj, 1 j 5 K , were Poisson processes with average intensities A,, 1 j K , mutually independent and indepenK , are dent of 8. The . latter independence property is the strong Markov property of multivariate Poisson processes. ). Also, ince a is non-negative, l(o,ll(T,) is fa+^,,-measurable.

This result contains most of the PASTA (Poisson Arrivals See Time Averages) and related results of queueing theory (see Chapter 3, fj 3). 1) Let N be a point process and {Ft)'be a history of N , both N and {F,)being compatible with the flow {Bt). Suppose that the intensity X o f N is finite and non-null, and let Pi be the Palm probability associated with (N, P ) . Suppose moreover that the Ft-predictable structure is adapted to (0,). jm derivative on ToToof P i with respect to P: On the other hand, the generalized Campbell's formula gives Combining all the above relations, we obtain where { X ( t ) ) is any arbitrary non-negative Ft-predictable process.

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