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Markov chains The prominent result of the analysis is the product-form solution for the oint 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 . generate barcode in asp.net using c# ASP . 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It is fully customizable and support for all barcode ... the rate out of a state due to a particular job is equal to the rate into that state due to that same job BCMP extension for the product-form solution The product form has been established under the assumption that each service station has the rst-come rst-served discipline and that the service times are exponentially distributed In an important paper of Baskett et al (1975) it has been shown that the product-form solution (5610) also holds when each service station uses one of the following four service disciplines, or BCMP disciplines: 1 The service discipline is rst-come- rst-served and the service times of the customers are exponentially distributed (multiple servers or state-dependent service is allowed) 2 The service discipline is processor-sharing; that is, if n customers are present at the station, each customer is served and receives service at a rate of 1/n The service time of a customer is allowed to have a general probability distribution 3 The service discipline is determined by an in nite number of servers; that is, each arriving customer gets immediately assigned a free server The service time of a customer is allowed to have a general probability distribution 4 The service discipline is pre-emptive resume, last-in rst-out; that is, customers are served one at a time in reverse order of arrival and a newly arriving customer gets immediate service, pre-empting anyone in service The service time of a customer is allowed to have a general probability distribution The product-form solution (5610) remains alid but the marginal probability distribution {pk (n), n = 0, 1, } of the number of customers present at station k depends on the service discipline at station k Under service discipline 1 with ck identical servers, the marginal distribution {pk (n)} is given by the equilibrium distribution of the number of customers present in the M/M/c queue with arrival rate = k , service rate = k and c = ck servers Under service discipline 3 at station k the marginal distribution {pk (n)} is given by the Poisson distribution with mean k E(Sk ), where the random variable Sk denotes the service time of a customer at each visit to station k Under both service discipline 2 and service discipline 4, at station k the marginal distribution {pk (n)} is given by the geometric n distribution {(1 k ) k , n = 0, 1, } with k = k E(Sk ), where Sk denotes the service time of a customer at each visit to station k 562 Closed Network Model In the performance evaluation of computer systems and exible manufacturing systems it is often more convenient to consider a closed network with a xed number of customers (jobs) A job may leave the system but is then immediately replaced by a new one The basic closed Jackson network is as follows: The network consists of K service stations numbered as j = 1, , K. 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