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Part Three
Planning and control
Types of queuing system
Conventionally queuing systems are characterized by four parameters.
M/M/m queues
G/G/m queues
A – the distribution of arrival times (or more properly interarrival times, the elapsed
times between arrivals)
B – the distribution of process times
m – the number of servers at each station
b – the maximum number of items allowed in the system.
The most common distributions used to describe A or B are either
(a) the exponential (or Markovian) distribution denoted by M; or
(b) the general (for example normal) distribution denoted by G.
So, for example, an M/G/1/5 queuing system would indicate a system with exponentially
distributed arrivals, process times described by a general distribution such as a normal distribution,
with one server and a maximum number of items allowed in the system of 5. This
type of notation is called Kendall’s notation.
Queuing theory can help us investigate any type of queuing system, but in order to
simplify the mathematics, we shall here deal only with the two most common situations.
Namely,
● M/M/m – the exponential arrival and processing times with m servers and no maximum
limit to the queue.
● G/G/m – general arrival and processing distributions with m servers and no limit to the
queue.
And first we will start by looking at the simple case when m = 1.
For M/M/1 queuing systems
The formulae for this type of system are as follows.
Using Little’s law,
Then,
WIP =
WIP = cycle time × throughput time
Throughput time = WIP / cycle time
u 1 t
Throughput time = × = e
1 − u 1 − u
and since, throughput time in the queue = total throughput time − average processing time,
W q = W − t e
u
1 − u
r a
t e
= − t e
1 − u
t
= e − t e (1 − u)
=
1 − u
u
= t e
(1 − u)
t e − t e − ut e
1 − u