# Other Notation

This page introduces some of the standard notation and symbols used when modelling problems using queueing theory. Not all of the notation on this page may necessarily by used in this project, but it may serve as useful reference material for the reader when reading other sources. The following table describes notation which is universally applicable.

Notation Meaning of Notation Units used

t Time since modelling began (). Any units can be used to represent time, as long as they are used consistently - in this section we represent the units of time with T.
State of the system Number of customers in the queueing system. Unitless.
Queue length Number of customers waiting for service. Unitless.
N(t) Number of customers in queueing system at time t. Unitless.
Pt(i) Probability of exactly i customers in queueing system at time t, given number at time 0. Note: One must be wary of this notation. In the author's experience, some sources will swap the t and i. Unitless.
s Number of servers in queueing system. Unitless.
Mean arrival rate (expected number of arrivals per unit time) of new customers when n customers are in the system. When the population size is infinite, the probability that another customer will want to join the queue is a constant, however full the queue is. When this is the case, the mean arrival rate can be denoted simply by the symbol  rather than the symbol . T-1
Mean service rate (expected number of customers completing service per unit time) when n customers are in the system. This represents the combined rate of all the servers in a multi-server environment. When the mean service rate per busy server is a constant independent of the number of customers in the system (as long as there is at least one customer), then it can be denoted by . In this case, provided  (in other words, all s servers are busy), . T-1
This is frequently termed the 'traffic intensity'. It is calculated with the simple formula:

One must be careful with this notation. Some sources use it to mean different things.

Erlang (named after A. K. Erlang, one of the founders of queueing theory).

The notation in the following table is also used, but it assumes that the queueing system is in a steady state. A steady state occurs when a queueing system has been in operation for a 'long' period of time. What length this is depends upon the particular situation. When the queueing system is first started, it is said to be in a transient state. One can say that the system has left the transient state and entered the steady state when the state of the system has become independent of the initial state and the elapsed time (this does not happen in some unusual circumstances).

Note: Below, the notation E(x) means the expected value of x. For more information on expected values, see the information on statistics for queueing theory.

Notation Meaning of Notation Units used

P(i) Probability of exactly i customers in the queueing system. See note above about notation Pt(i). Unitless.
L Expected number of customers in the queueing system. Unitless.
Lq Expected queue length (excludes customers being served). Unitless.
Waiting time in system (includes service time) for each customer. Unitless.
W E() Unitless.
Waiting time in queue (excludes service time) for each customer. Unitless.
Wq E() Unitless.

Last Updated: 14th June 1999
Written by: Andrew Ferrier