12.4. 12.4. The  Queue | ||
|---|---|---|
| Prev | Chapter 12. Finite-Source Systems | Next |
Consider the homogeneous finite-source model with
, 
, independent servers. Denoting by 
the number of customers in the system at time
similarly to the previous sections it can easily
be seen that it is a birth-death process rates
with intensity.
The steady-state distribution can be obtained as
with normalizing condition
To determine 
we can use the following simpler recursion.
Let
and using the relation for the consecutive elements of the birth-death process our procedure operates as follows
Since
must be satisfied thus we get
Dividing both sides by 
we have
hence
Finally
Let us determine the main performance measures
1. Mean number of customers in the systems can be computed as
2. Mean queue length can be obtained by
3. Mean number of customers in the source can be calculated by
4. Utilization of the system is computed by
5. Mean busy period of the systems can be obtained by
6. Mean number of busy servers can be calculated by
Furthermore
7. Mean number of idle servers
Additional relation is
8.Utilization of the sources can be calculated by
9. The mean waiting and response times can be derived by
thus for the mean waiting time we have
Hence the mean response time is
consequently we get
which is the well-known Little's formula. Thus we get
that is
Show that
because from this follows
which is the Little's formula for the waiting time.
Since
where
Furthermore, it is well-known that
We can proceed as
Finally we get
or in another form
that is
| mean arrival rate = mean service rate, |
which was expected because the system is in steady state. Consequently
10. Mean idle period of a server can be computed as follows.
If the idle servers start their busy period in the order as they
finished the previous busy period, then their activity can be written
as follows. If a server becomes idle and finds other 
servers idle, then his busy period start at the
instant of the arrival of the 
th customer.
Let 
denote the mean idle period of the server and let
denote the mean conditional idle period mention
above. Clearly 
Let 
can be computed by the help of the theorem of
total expectation, namely
where
is the probability that there is an idle server.
11. Mean busy period of the server can be calculated as follows.
Since
thus
That is
This subsection is devoted to the most complicated problem of this system, namely to the determination of the distribution function of the waiting and response times. First the density function is calculated and then we obtain the distribution function. You may remember that the distribution has been given in the form
Introducing 
, this can be written as
thus
It is easy to see that the probability of waiting is
Inserting 
this can be rewritten as
We show that the distribution function of the waiting time can be calculated as
and thus
which is probability that an arriving customer finds idle server. For the density function we have
If we calculate the integral 
that is 
is not considered then
By the substitution 
we have 
for the integral part we get
that is
as it was expected. Thus
Let us determine the density function for 
. That is
as we got earlier, but we have to remember that
Therefore
Thus for the distribution function we have
which was obtained earlier.
To verify the correctness of the formula let 
. After substitution we get
but
thus
The derivation of the distribution function of the response time is analogous. Because the calculation is rather lengthly it is omitted, but can be found in the Solution Manual for Kobayash [ 50 ].
As it can be seen in Allen [
2
], Kobayashi [
50
], the following formulas are valid
for 
where
Hence the density function can be obtained as
It should be noted that for the normalizing constant we have the following recursion
with initial value
First determine the Laplace-transform of the waiting time.
It is easy to see that by using the theorem of total Laplace-transform we have
We calculate this formula step-by-step. Namely we can proceed as
Then
Then 
. Thus the last equation can be written as
Finally collecting all terms we get
To verify the correctness of the formula let 
.
Thus after inserting we have
as we got earlier.
Keeping in mind the relation between the waiting time and the response time and the properties of the Laplace-transform we have
which is in the case of 
reduces to
.
| Java applets for direct calculations can be found at |
Example 12.3. A
factory possesses 
machines having mean lifetime of 
hours. The mean repair time is 
hours and the repairs are carried out by
repairmen. Find the performance measures of the
system.
Solution:
By using the recursive approach we get
and so on.
Hence
From here
The distribution can be seen in the next Table for
, 
, 
Hence the performance measures are
Let us compare these measures to the system where we have
machines and a single repairman. The lifetime
and reapir time characteristcs remain the same. The result can be
seen in the next Table
Example 12.4. Let
us continue the previous Example with cost structure. Assume that
the waiting cost is 
Euro/hour and the cost for an idle repairman is
Euro/hour. Find the optimal number of repairmen.
It should be noted that different cost functions can be
constructed.
Solution:
The mean cost per hour as a function of 
an be seen in the next Table which are
calculated by the help of the distribution listed below for
The mean cost per hour is
Hence the optimal number is 
.
This simple Example shows us that there are different criteria for the optimal operation.
Queue