Page 222 - A First Course In Stochastic Models

QUEUEING NETWORKS 215

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 J.R. Jackson (1957,
1963), who discovered the product-form solution. In Section 5.6.1 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 fixed number of customers move around in
the network. The closed network will be analysed in Section 5.6.2. For clarity
of presentation the analysis is restricted to a single class of customers. In appli-
cations, 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.

5.6.1 Open Network Model
As a prelude to the open queueing network model, consider the following medi-
cal 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 labo-
ratory 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 inde-
pendent Poisson processes with respective rates r 1 , . . . , r K .

• 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/µ j for j = 1, . . . , K.

• Upon service completion at station i, the served customer moves with probability
p ij to station j for j = 1, . . . , K or leaves the system with probability p i0 =
K
1 − j=1 p ij .