This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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 ....
View
Full Document
 Spring '09
 all
 Queueing theory, J. Virtamo, queueing networks

Click to edit the document details