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Queueing network models are a useful analysis tool in a wide variety of areas such as computer performance evaluation, communication network design and production planning in exible manufacturing Generally speaking, a network of queues is a collection of service nodes with customers (jobs) moving between the nodes and making random requests for service at the nodes Under appropriate conditions these networks can be modelled and analysed by means of continuous-time

.

Markov chains The prominent result of the analysis is the product-form solution for the joint distribution of the numbers of customers present at the various nodes Networks that can be described by a continuous-time Markov chain and have the product-form solution are often called Jackson networks after JR Jackson (1957, 1963), who discovered the product-form solution In Section 561 we consider the open network model A network is called open if external arrivals occur at one or more nodes and departures from the system occur at one or more nodes A network is called closed when a xed number of customers move around in the network The closed network will be analysed in Section 562 For clarity of presentation the analysis is restricted to a single class of customers In applications, however, one often encounters networks of queues with several customer classes The results presented in this section can be extended to the case of multiple customer classes 561 Open Network Model As a prelude to the open queueing network model, consider the following medical application involving the analysis of emergency facilities Patients arrive at an emergency room for late-night operations Incoming patients are initially screened to determine their level of severity On average, 10% of incoming patients require hospital admission Twenty percent of incoming patients are sent to the ambulatory unit, 30% to the X-ray unit and 40% to the laboratory unit Patients sent to the ambulatory unit are released after having received ambulatory care Of those going to the X-ray unit, 25% require admission to the hospital, 20% are sent to the laboratory unit for additional testing, and 55% have no need of additional care and are thus released Of patients entering the laboratory unit, 15% require hospitalization and 85% are released This emergency system provides an example of a network of queues Consider now the following model for an open network of queues (open Jackson network ): The network consists of K service stations numbered as j = 1, , K External arrivals of new customers occur at stations 1, , K according to independent Poisson processes with respective rates r1 , , rK Each station is a single-server station with ample waiting room and at each station service is in order of arrival The service times of the customers at the different visits to the stations are independent of each other, and the service time of a customer at each visit to station j has an exponential distribution with mean 1/ j for j = 1, , K Upon service completion at station i, the served customer moves with probability pij to station j for j = 1, , K or leaves the system with probability pi0 = K 1 j =1 pij.

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Detection points for the PDP context state model Arming mode (scenario 1) EDP-N, EDP-R Arming mode (scenario 2) TDP-R Description

The constant gives the long-run average arrival rate of customers The limit exists when customers arrive according to a renewal process (or batches of customers arrive according to a renewal process with independent and identically distributed batch sizes) The existence of the above limits is suf cient to prove the basic relations Lq = Wq and L = W (231) (232)

These basic relations are the most familiar form of the formula of Little The reader is referred to Stidham (1974) and Wolff (1989) for a rigorous proof of the formula of Little Here we will be content to demonstrate the plausibility of this result The

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