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Extra resources for A Boundary Value Problem for a Second-Order Singular Elliptic Equation in a Sector on the Plane

Example text

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.