1. DECISION MODELING WITH Prentice Hall Publishers and
MICROSOFT EXCEL
Chapter 15
QUEUING
Part 2
Copyright 2001
Prentice Hall Publishers and
Ardith E. Baker

2. MODEL 2: A FINITE QUEUE (WATS LINES)
In this model, we attempt to select the appropriate number of _______lines for St. Luke’s.
The telephone company can provide a great deal of _______in these matters, since queuing models have found extensive use in the field of _________ traffic engineering.
This problem is typically attacked by using the _______model, “with blocked customers cleared.”
This is a _____________queue with s servers, exponential interarrival times for the calls and a general distribution for the ______________(length of call).

3. “Blocked customers cleared” means that when an _______finds all of the servers __________(all of the lines busy), he or she does not get in a queue but simply________.
Probability of j Busy Servers The problem of selecting the appropriate number of _____(servers) is attacked by computing the steady-state ______________that exactly j lines will be busy.
This will be used to calculate the steady-state probability that all s _______are busy.
Clearly, if you have s lines and they are all busy, the next _________will not be able to place a call.

4. The steady-state probability that there are exactly j busy ________ given that s lines (servers) are available is:
Pj =
(l/m)j /j! S
k=0 s
(l/m)k /k!
where l = _______rate (the rate at which calls arrive)
1/m = mean _______time (the average length of a conversation)
s = number of _________(lines)
This expression is called the ____________Poisson distribution or the ________loss distribution.
The value of Pj depends only on the _______of this distribution.

5. S Consider a system in which
l = 1 (calls arrive at the rate of 1 per minute)
1/m = 10 (the average length of a conversation is 10 minutes)
Here, l/m = 10. Suppose there are five lines in the system (s = 5) and we want to find the steady-state probability that exactly two are busy (j = 2).
P2 =
(l/m)2 /2! S
k=0 5
(l/m)k /k!
P2 =
(10)2 /2•1
/ /2• /3•2• /4•3•2• /5•4•3•2•1
On the average, two lines would be busy 3.4% of the time.
P2 = 50
= 0.034

6. An ___________way of obtaining Pj that is easy to implement in a spreadsheet is as follows:
Pi = Pi-1 (l/m)/i
So, for example, once we know P2, we can ________ P3 as:
P3 = P2(10)/3
= (0.034)(10)/3 =
Likewise, P4 is found as:
P4 = P3(10)/4
= (0.1133)(10)/4 =
Each _________Pi-1 is multiplied by (l/m) and divided by i to achieve the new Pi.

7. The more interesting question is: “What is the ___________that all of the lines are busy?” since in this case, a potential caller would not be able to place a call on the _______lines.
To find the answer to this question, we simply set ______(in our example s = 5) and we obtain
P5 = P4(10)/5
= (0.2833)(10)/5
= 0.564
Or on the average the system is totally _________ 56.4% of the time.

8. Using the spreadsheet, the probability that a customer _______with 5 servers is easily calculated.
We can then build a data table to determine this probability for different _________of s.

9. Enter the values for s, ranging from 0 to 10.
Specify the formula (= F13) for the quantity that we want to track (the prob. that a customer balks).
Highlight the cell range B22:C33 and click Data – Table.
Specify $E$4 as the Column Input and click OK.

10. Column D calculates the __________improvement in this probability as servers are added.
It is clear that the marginal effect of adding more servers__________.

11. Average Number of Busy Servers This quantity is called the__________
Average Number of Busy Servers This quantity is called the__________. If we define N as the average number of busy servers, then N
= (l/m)(1 – Prob. that a customer will balk)
Assume for this model, l = 1 and 1/m = 10.
Thus, if 10 lines are_________, the probability that all 10 are busy is (from previous table).
It follows then that N
= (10)(1 – 0.215) = 7.85
After finding N, the server ____________can be calculated by dividing N by s (the number of servers). Thus, 7.85/10 = 78.5%. Each server is busy 78.5% of the time and ______21.5% of the time.

12. MODEL 3: THE REPAIRPERSON MODEL
Now we must decide how many repairpersons to hire to _________20 pieces of electronic equipment.
Machines are _________on a first-come, first-served basis and a _______repairperson treats each broken machine.
The failed machines form a ________in front of the multiple servers (repairpersons).
This is an ______model, but it differs from the blood-testing model in that there is a _______number of items (20) that can join the queue.

13. Queuing models in which only a ________number of “people” are eligible to join the queue is said to have a finite_______________________.
Models with an ______________number of possible participants are said to have an __________calling population.
Consider the model with 20 machines and 2 repair-persons. Assume that
when a machine is running, the time between ______________has an exponential distribution with parameter l = 0.25 per hour.
Thus, the average ____between breakdowns is 1/l = 4 hours.

14. The time it takes to ______a machine has an exponential distribution and the _______repair time (1/m) is 0.50 hour.
This model is an M/M/2 model with a ____________of 18 items in the ______and a finite calling population.
In this case, the general equations for the steady-state probability that there are _______in the system is a function of l, m, s, and N (the number of _____________).
Pn = N!
n!(N – n)!
(l/m)n P0
for 0 < n < s
for s < n < N
Pn = N!
(N – n)!s!sn-s
(l/m)n P0

15. S
n=0 N
Pn = 1
We also know that
We thus have N + 1 ________equations in the N + 1 variables of interest (P0, P1, …, Pn).
Although_____________, this makes it possible to calculate values of Pn for any particular model.
There are, however, no simple ____________for the expected number of jobs (broken machines) in the system or for___________.
If the values for Pn are computed, then it is a simple task to find a _____________value for the expected number in the system. You must just calculate: S
n=0 N
nPn
expected number in system = L =

16. A spreadsheet can be used to compute values of Pn.
NOTE: when you enter the value for the arrival rate (l) in the “MMs” worksheet, you need to enter N*l (the entire population’s arrival rate).

17. TRANSIENT vs STEADY-STATE RESULTS: ORDER PROMISING
In this section, we will consider a situation in which we are interested in the _________(not steady-state) behavior of the system.
_____________processes can be viewed as complex queuing systems.
In fact, queuing systems __________is probably the most frequently used management science tool in manufacturing.
SONOROLA company is concerned about when to ________a new customer order. The order is for 20 units of an item that requires __________processing at 2 work stations.

18. (10 units x 4 hrs/unit + 4 hrs + 4 hrs)  8 hrs/day = 10.5 days
The average _______to process a unit at each work station is 4 hours. Each work station is ________for 8 hours every working day.
By considering when the last of the 20 units will be ___________, it is estimated that it will take 10.5 days to process the order.
The last unit must wait at Work Station 1 for the first 19 units to be completed, then it must be _________at Work Station 1, then at Work Station 2.
Assuming that it does not have to wait when it gets to Work station 2, we can calculate the following:
(10 units x 4 hrs/unit + 4 hrs + 4 hrs)  8 hrs/day = 10.5 days
However, this analysis is somewhat___________. It ignores the ___________of the processing times and the possibility of queuing at Work Station 2.

19. The 4-hour _____________time at each work station was arrived at by __________many processing times that were less than 4 hours with a few processing times that were significantly ________than 4 hours (due to equipment failures at a work station while processing a unit).
Next, check to see whether the _____________of the basic queuing model are met.
The output from Work Station 1 are the _________to Work Station 2, and the time between arrivals is _____________because the processing time at Work Station 1 is exponential.
The service time at Work Station 2 is ___________ because it is the same as the ______________time.

20. The units are processed on a_________, first-served basis at Work Station 2 and there is sufficient _____ capacity between the work stations so that the queue size is_________.
However, the __________of an infinite time horizon is not met. We are only interested in the ________of the system until “________” 20 ends its processing.
Let’s apply the basic model anyway and use it as an _________________.
The time it takes to process 20 units is approximated as follows:
1. The last unit in the batch of 20 is estimated to leave Work Station 1 after 20 x 4 = 80 hours.

21. 2. This unit will then wait in the _______in front of Work Station 2.
3. Finally, it will ________processing at Work Station 2, at which time all 20 units will have been completed.
The total time that the last unit spends at Work Station 2 is W. Thus, our __________is 20 x 4 x W.
Remember, for the basic model, W = 1/(m – l), for m > l.
The problem is that m and l are ______(1 unit per 4 hours).

22. A spreadsheet can be used to simulate the ______of the 20 units through the 2 work stations.
Assume that raw material is always ____________at Work Station 1 so that the next unit at Work station 1 can start as soon as the ___________unit is finished.
This means that for Work Station 1, the start time of a unit is the __________of the previous unit.
The start time of a unit at Work station 2 is either the stop time of ________on Work Station 1 or the stop time of the previous unit on Work Station 2, whichever is________.
The stop time of a unit is just the ____________plus the _____________time. The finish time in days is calculated by dividing the stop time at Work Station 2 of the last unit by the number of___________.

23. The finish time is calculated to be 10
The finish time is calculated to be 10.5 days if every unit takes exactly 4 hours at every work station.

24. To analyze the impact of __________time variability, replace the _________processing time of 4 at Work Station 1 in the spreadsheet with the appropriate ________distribution (exponential with a mean of 4).
enter =RiskExpon($B$1) to make the _______time a random variable.
We would like to know the 99th ___________of this random variable so that we could then promise the order in that number of days and be _____sure that it would actually be _____________on time.

25. The output shows the 99th percentile for cell F2 (the finish time in days) based on 1000 sets of 40 random processing times.

26. To be 99% sure of having the order completed by the _________date, set the due date to be days after the material becomes ____________at Work Station 1.
The ____________that takes place at Work Station 2 has increased the _________(time from the start of the order to its completion) by nearly 8 days (18.28 – 10.5) over what it would be if there were no __________in the processing times.

27. Here is a histogram of the finish time
Here is a histogram of the finish time. Note that the time can vary from 6.5 days up to 21 days.

28. THE ROLE OF THE EXPONENTIAL DISTRIBUTION
The role of the ___________distribution in ________ queuing models is useful in understanding the use of queuing models.
Most analytic results for queuing situations involve the exponential distribution either as the distribution of ___________times or service times or both.
The following three properties help to identify the set of _______________in which it is reasonable to assume that an exponential distribution will______.

29. 1. _______________: In an arrival process, this property implies that the ____________that an ________will occur in the next few minutes is not influenced by when the last arrival occurred.
This situation arises when
(a) there are many ___________who could potentially arrive at the system
(b) each person decides to arrive _____________of the other individuals
(c) each individual selects his or her time of arrival completely at_________

30. 2. ___________________: With an exponential distribution, small values of the ________time are common (as shown below).
Prob S<t
1.0
0.632 10 20 30 40 t
This graph shows the _____________that the service time S is less than or equal to t if the ________service time is 10.

31. The graph showed that more than 63% of the service times were _________than the average service time (10).
Compare this to the ___________distribution where only 50% of the service times are ___________than the average.
The practical implication is that an exponential distribution can best be used to model the distribution of _______________in a system in which a large proportion of “jobs” take a very ___________and only a few “jobs” run for a long time.

32. 3. Relation to the __________________: While introducing the_____________, a relationship between the exponential and Poisson distributions was noted.
In particular, if the time between arrivals has an ___________distribution with parameter l, then in a specified period of time (say, T) the number of arrivals will have a ___________ distribution with parameter lT.
Then, if X is the number of arrivals during the time T, the probability that X equals a specific number (say, n) is given by the equation
Prob [X = n] =
e-lT(lT)n n!
For any _____________integer value of n.

33. The _____________between the exponential and the Poisson distributions plays an important role in the theoretical ______________of queuing theory.
It also has an important practical________________.
By comparing the number of _______that arrive for service during a specific period of time with the number that the Poisson distribution_____________, the manager is able to see if his or her choices of a model and ____________values for the arrival process are reasonable.

34. QUEUE DISCIPLINE
In addition to the _________distribution, service distribution and number of servers, the queue __________must also be specified to define a queuing system.
So far, we have always assumed that arrivals were served on a first-come, first-serve basis (often called __________, for “first-in, first-out”).
However, this may not always be the case. For example, in an elevator, the last person in is often the first out (___________).
Adding the possibility of selecting a ________queue discipline makes the queuing models more ____________.
These models are referred to as __________models.