
A situation familiar to everyone is waiting in a line. A typical
example might be the line of customers that forms in front of
the service windows at a post office. The number of customers
in the line grows and shrinks with time, and, as anyone who
has had the experience knows, the wait can be highly unpredictable.
Because the number in line is a random variable that changes
with time, the system of customers and servers fits the definition
of a stochastic process.
Other familiar situations where lines or queues form include
a ticket booth at a theater, a conference registration desk,
a red light at a traffic signal, buffer storage on an assembly
line, email on a server, and taxis outside an airport. Basically,
a queue results whenever existing demand temporarily exceeds
the capacity of the service facility; i.e., whenever an arriving
customer cannot receive immediate attention because all servers
are busy. This situation is almost always guaranteed to occur
at some time in any system that has probabilistic arrival and
service patterns. Tradeoffs between the cost of increasing service
capacity and the cost of waiting customers prevent an easy solution
to the design problem. If the cost of expanding a service facility
were no object, then theoretically, enough servers could be
provided to handle all arriving customers without delay. In
reality, though, a reduction in the service capacity results
in a concurrent increase in the cost associated with waiting.
The basic objective in most queuing models is to achieve a
balance between these costs.
In this section, we are primarily concerned with continuoustime
systems operating in steady state. A variety of analytical results
are available for these systems mostly under the assumption
that the arrival and service processes are Markovian. Formulas
embodied in the Queuing addin compute statistical estimates
for such measures as the average number in the queue, the average
waiting time for a customer, and the probability that the service
mechanism is busy. We do not repeat many formulas in this section,
but concentrate on describing the models for which the formulas
are available.
One of the key insights gained from studying queuing systems
is that they may not be very efficient in terms of resource
utilization. Queues form and customers wait even though servers
may be idle much of the time. The fault is not in the model
or underlying assumptions. It is a direct consequence of the
variability of the arrival and service processes. If variability
could be eliminated, systems could be designed economically
so that there would be little or no waiting, and hence no need
for queuing models.
