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Unformatted text preview: J. Virtamo 38.3143 Queueing Theory / Queueing networks 1 QUEUEING NETWORKS A network consisting of several interconnected queues • Network of queues Examples • Customers go form one queue to another in post office, bank, supermarket etc • Data packets traverse a network moving from a queue in a router to the queue in another router History • Burke’s theorem, Burke (1957), Reich (1957) • Jackson (1957, 1963): open queueing networks, product form solution • Gordon and Newell (1967): closed queueing networks • Baskett, Chandy, Muntz, Palacios (1975): generalizations of the types of queues • Reiser and Lavenberg (1980, 1982): mean value analysis, MVA J. Virtamo 38.3143 Queueing Theory / Queueing networks 2 Jackson’s queueing network (open queueing network) Jackson’s open queueing network consists of M nodes (queues) with the following assumptions: • Node i is a FIFO queue – unlimited number of waiting places (infinite queue) • Service time in the queue obeys the distribution Exp( μ i ) – in each queue, the service time of the customer is drawn independent of the service times in other queues – note: in a packet network the sending time of a packet, in reality, is the same in all queues (or differs by a constant factor, the inverse of the line speed) – this dependence, however, does not markedly affect the behaviour of the system (so called Kleinrock’s independence assumption) • Upon departure from queue i , the customer chooses the next queue j randomly with the probability q i,j or exits the network with the probability q i,d (probabilistic routing) – the model can be extended to cover the case of predetermined routes (route pinning) • The network is open to arrivals from outside of the network (source) – from the source s customers arrive as a Poisson stream with intensity λ – fraction q s,i of them enter queue i (intensity λq s,i ) J. Virtamo 38.3143 Queueing Theory / Queueing networks 3 Node i in Jackson’s network s = source, external d = destination, sink N i = number of customers in queue i l i l q s, i l q d i i, l q 1, i 1 l q 1 i i, l q M, i M l q M i i, l i m i Exp( ) m i N i . . . . . . Without complications one could assume state dependent service rates μ i = μ i ( N i ). This could describe e.g. multiserver nodes. To simplify the notation, we assume in the sequel a constant sevice rate μ i . Jackson’s network The opnenness of the network requires that from each node there is at leas one path ( negationslash = 0) to the sink d , i.e. the probability that a customer entering the network will ultimately exit the network is 1. l 1 m 1 l 2 m 2 l 3 m 3 l 4 m 4 nielu d l l l q s, 1 l q s, 2 l q s, 3 l q 3, 2 3 l q 3, 4 3 l q 2, 4 2 l q 2, d 2 l q 1, 2 1 l q 1, d 1 l q 1, 4 1 l q 4, 1 4 lähde s J. Virtamo 38.3143 Queueing Theory / Queueing networks 4 Conservation of the flows Denote λ i = average customer flow through node i ....
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This note was uploaded on 01/05/2010 for the course ECE 01 taught by Professor All during the Spring '09 term at Aarhus Universitet.
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