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Part Three
Planning and control
Variability
The concept of variability is central to understanding the behaviour of queues. If there were
no variability there would be no need for queues to occur because the capacity of a process
could be relatively easily adjusted to match demand. For example, suppose one member of
staff (a server) serves at a bank counter customers who always arrive exactly every five minutes
(i.e. 12 per hour). Also suppose that every customer takes exactly five minutes to be served,
then because,
(a) the arrival rate is ≤ processing rate, and
(b) there is no variation
no customer need ever wait because the next customer will arrive when, or before, the
previous customer. That is, WIP q = 0.
Also, in this case, the server is working all the time, again because exactly as one customer
leaves the next one is arriving. That is, u = 1.
Even with more than one server, the same may apply. For example, if the arrival time at
the counter is five minutes (12 per hour) and the processing time for each customer is now
always exactly 10 minutes, the counter would need two servers, and because,
(a) arrival rate is ≤ processing rate m, and
(b) there is no variation
again, WIP q = 0, and u = 1.
Of course, it is convenient (but unusual) if arrival rate/processing rate = a whole number.
When this is not the case (for this simple example with no variation),
Utilization = processing rate/(arrival rate multiplied by m)
For example, if arrival rate, r a = 5 minutes
processing rate, r e = 8 minutes
number of servers, m = 2
then, utilization, u = 8 / (5 × 2) = 0.8 or 80%
Incorporating variability
The previous examples were not realistic because the assumption of no variation in arrival or
processing times very rarely occurs. We can calculate the average or mean arrival and process
times but we also need to take into account the variation around these means. To do that we
need to use a probability distribution. Figure S11.1 contrasts two processes with different
arrival distributions. The units arriving are shown as people, but they could be jobs arriving
at a machine, trucks needing servicing, or any other uncertain event. The top example shows
low variation in arrival time where customers arrive in a relatively predictable manner. The
bottom example has the same average number of customer arriving but this time they arrive
unpredictably with sometimes long gaps between arrivals and at other times two or three
customers arriving close together. Of course, we could do a similar analysis to describe processing
times. Again, some would have low variation, some higher variation and others be
somewhere in between.
In Figure S11.1 high arrival variation has a distribution with a wider spread (called
‘dispersion’) than the distribution describing lower variability. Statistically the usual measure
for indicating the spread of a distribution is its standard deviation, σ. But variation does not
only depend on standard deviation. For example, a distribution of arrival times may have
a standard deviation of 2 minutes. This could indicate very little variation when the average
arrival time is 60 minutes. But it would mean a very high degree of variation when the