By Maury Bramson
Queueing networks represent a wide family members of stochastic versions, concerning jobs that input a community, compete for provider, and finally depart the community upon of completion of provider. because the early Nineteen Nineties, big consciousness has been dedicated to the query of while such networks are solid.
This quantity provides a precis of such paintings. Emphasis is put on using fluid types in exhibiting balance, and on examples of queueing networks which are risky even if the arriving expense is lower than the carrier rate.
The fabric of this quantity is predicated on a sequence of 9 lectures given on the Saint-Flour likelihood summer time tuition 2006.
Lectures have been additionally given by means of Alice Guionnet and Steffen Lauritzen.
Read or Download Stability of Queueing Networks: École d'Été de Probabilités de Saint-Flour XXXVI - 2006 PDF
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Additional resources for Stability of Queueing Networks: École d'Été de Probabilités de Saint-Flour XXXVI - 2006
N, with probability β(i, n), where n is the number of jobs in the node including this job. This function is the same as that given in (c). 5, we will also be interested in symmetric queueing networks. These networks are deﬁned analogously, with properties (a)-(d) being assumed to hold at each station, and departing jobs from a class k being routed to a class with probability Pk, . 3 for homogeneous networks, where the procedure is the same. 3, that δ = β, in parts (c) and (d). These properties are more general in that the service time means mk need no longer be equal at diﬀerent classes.
K}. 6). We will employ notation similar to what was used earlier in the section, with Ak (x), Ek (x), Ij (x), and Rk, (x) denoting the states y obtained from x by the diﬀerent types of transitions. As before, Ak (x), Ek (x), and Ij (x) will denote the states obtained by an arrival into the network at k, an exit from the network at k, and an internal state change at j. For y ∈ Ak (x) or y ∈ Ek (x), we will require that y j = xj for j = s(k), and that the number of jobs at k increase or decrease by 1, and elsewhere remain the same.
Of course, n n i=1 β(i, n) = i=1 δ(i, n) = 1. We have emphasized in the above deﬁnition that the external arrival rates and service rates β and δ do not depend on the class of the job. 7, the main result in this section. 4 for symmetric nodes, as will be our assumption that the interarrival times are exponentially distributed. On the other hand, the assumptions that the service times be exponential and that their means mk be constant, which are needed in this section, are not needed for symmetric nodes.