for Q_Sample provides answers to the questions originally
posed. How many machines on the average will be waiting for
repair? This is given by the mean number in the queue or
3.51. The mean number in service (or actually being repaired)
is 2.50, so the total number in the system is 6.01. This
is an interesting number because it represents the average
inventory due to the repair operation.
How much time will a machine spend in
the repair facility? This is given by the mean time in the
system as 1.20 hours. The time dimensions are the same as
those assumed for the arrival and service rates. The system
time is broken into time in the queue (0.70 hours) and time
in service (0.50) hours. These numbers are interesting because
they describe the cycle time for the repair process.
How often will the workers be idle?
The efficiency result indicates that on the average the repair
workers are busy 83.3% of the time. The state probabilities
indicate that all three workers are idle simultaneously 4.5%
of the time (P(0)), all three are busy 70% of the time (1
- P(0) - P(1) - P(2)), and the system is never full since
in this case there is no limit to the length of the queue.
The entry for P(Wait >= Critical
Wait) shows the probability that the time in the queue
is greater than a specified value. In the display the critical
value is 0.5 hours.
The column for Q_2 shows the results
for a finite queue case. With a finite queue there is the
possibility that an arrival will discover that the system
is full. For the example this occurs when there are 8 already
in the system. When the system is full, the arrival balks and
does enter. The probability of this is P(8) or 0.0615. This
is an important measure for a system because it indicates
the proportion not served. It also appears as the Probability
that the system is full. When balks occur the Throughput
Rate is smaller than the arrival rate, as indicated
for the example.
Column Q_3 illustrates a finite population system. In this
case the arrival rate is the rate for each individual of the
population. The actual arrival rate into the system depends
on the state since customers already in the system cannot arrive.
For the example there are 8 in the population, so the arrival
rate into the system when it is empty (state 0) is 0.0625*8
= 5. Since K and N are equal in this case there
is no balking. When there are 8 customers in the system, none
can arrive. The Throughput Rate for the finite population
case is smaller that the other two. This is due to reduction
in arrival rate as the number in the system increases.