Assumptions
In modeling service systems, we'll make three simplifying assumptions.
The arrival rate is constant, and arrivals are described by
a Poisson distribution with parameter l. This is just a fancy
way of saying that customer arrivals are independent of each
other.
The service rate is constant, and the distribution of service
completion times is described by an exponential distribution
with parameter m.
The system is in equilibrium — that is, the doors to the
facility opened a long time ago. We have to assume this because
just after the doors open, there is nobody in line, and it takes
a while for things to settle, which makes it a lot more difficult
to describe.
From these assumptions, we'll derive relationships
for some important performance parameters of a service system.
These formulas are useful in decision-making regarding capacity
utilization for the system. The quantities that are most useful
in describing the system are.
- Ls: the average number of customers in the system,
either waiting in line or being serviced.
- Ws: the total elapsed time to get through the
system, including both waiting time and service time.
- Lq: the average number of customers in the queue.
- Wq: the average elapsed time spent waiting in the queue.
- Pn: the probability that there are n customers in the system.
- Pn>k: the probability that there are more than
k customers in the system.
- r: the average utilization rate
of the server. This is the fraction of the time that the server isn't idle.
- La: the average length of non-empty lines.
- Wa: the average waiting time for customers who actually have to wait.
Single-server
systems
A most important relationship is the Fundamental Balance Equation, which is based on the equilibrium
assumption. If we let Pn be the probability that there
are n customers in the system, either waiting in line or being
served, then
lPn-1 = mPn
What does this mean? Just that since we're in equilibrium,
on average the number of customers in the system can't change.
Thus if there are n-1 customers, the probability that we go to
n in the next interval of time t is ltPn-1.
Similarly, if there are n customers in the system, the probability
that we finish one is mtPn.
These two probabilities must be equal if we want to stay in equilibrium:
ltPn-1 = mtPn
and we see that the factor of t on both sides can be eliminated.
We can use the Fundamental Balance Equation to derive the
average number of customers in the system. We'll do this by first
calculating the probability Pn that there are n customers
in the system. Let P0 be the probability that there
are no customers in the system and P1 be the probability
that there is one customer in the system. Then
and
Continuing in this way 
. Now
since
,
it follows that
Here we assumed that l < m, in order to close the sum of powers
of l/m. This sum is just a geometric
progression. Unless this assumption is valid, the sum of powers
is infinite. But this assumption is the only interesting case
anyway, because if m < l, the waiting
lines become infinite in length and never reach equilibrium.
Thus P0 = 1 - l/m and
Pn = (1 - l/m)(l/m)n
From this we can immediately derive r,
the utilization rate of the server. Since P0 is the
probability of zero customers in the system, it's also the probability
that the server is idle. Thus, the utilization rate r
of the server is just 1 - P0, so
r = 1 - P0 = l/m
Another performance measure of interest is the probability
that there are fewer than j customers in the system:
Thus
.
Now that we know the probability of there being n customers
in the system, it's relatively simple to figure out the average number of customers in the system.
We won't prove it, but this last form can be rewritten as
.
We can also derive a relationship between Ls and
Ws. Let's think about the state of the system just
after a customer has been serviced. On average there are then
Ls customers in the system, and they have all arrived
in the time it took that last customer to get through the system.
On average, the time it took that last customer to get through
the system was Ws, so the average number of new customers
that arrived in that time was lWs.
Thus
Ls = lWs
Since we have already have an expression for Ls
in terms of l and m,
we have
.
Let's now find expressions for the average length
of the queue, and for the time spent waiting in the queue. For
some businesses, these attributes of the system are critical
determiners of customer satisfaction (or dissatisfaction!), and
they also determine the size of the waiting facility.
Consider the state of the system just after a customer is
called to be serviced. On average there are then Lq
customers in the queue, and they have all arrived in the time
it took that last customer to be called by the server for service.
On average, the time it took that last customer to get through
the queue was Wq, so the average number of new customers
that arrived in that time was lWq.
Thus
Lq = lWq.
Since m is the average number of
customers serviced per unit time, the average time required to
service a customer is 1/m. And since
the average total time spent in the system is equal to the average
time spent waiting in line plus the average time spent being serviced,
Ws = Wq + 1/m.
Since we know how to express Ws in terms of l and m, it
follows that
and
.
To find the average length of non-empty lines, we divide the
average length of a queue by the probability that there is a queue at all:
The expected waiting time for those who actually wait is the
average time to service La customers:
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his page gives some explanations
of the formulas for service system performance parameters. You
don't need to read this to do the homework, but if you're curious
about where the formulas come from, read on.
