The figure below shows the queuing
system under consideration. Customers requiring some service
are the small circles and servers are the numbered rectangles.
Customers arrive to the system from some input source. If some
server is not busy, the customer immediately begins to be served.
Otherwise, the customer must wait in a queue until a server
is available. Some time is required for service, after which
the customer departs.
The input source, also called the calling population,
is the collection of customers providing inputs to the queuing
system. We assume for this model that the calling population
is infinite, that is, the rate of arrivals into the system
is unaffected by the number already there.
A queue discipline defines the rules by which
customers are selected for service. A common discipline, assumed
here, is first-come-first-served. Service is provided by one
or more servers (or channels) operating in parallel. The servers
may or not be identical.
The queuing system is the combination of the
queue and the service channels. For this model, we assume that
the total number of customers that can be present in the system
is limited to some maximum number. That is, the size of the
queue is finite and limited by the maximum number in the system
less the number of servers. When customers arrive and find
the queue to be full, the customer does not enter the system
and does not receive service.
The state/event model is illustrated by the network
below. The states are the nodes and the number in each circle
is the number of customers in the system. The event of an arrival
indicated by a, adds 1 to the number in the system.
The number of servers is s. When the state number
is less than or equal to s, all customers are in service
and there is no queue. When the number in the system exceeds s, a
queue forms.The service event, indicated by d, reduces
the number in the system.
The queuing model is really a birth-death model
with prescribed birth and death rates that depend on the number
in the system.