Problems 1. A bank has a drive-in window, which is open from 10 a.m. to 3 p.m. on business days. Customers drive up at a mean rate of 12 per hour, according to a Poisson distribution. The teller requires a mean of 2.4 minutes to serve each customer. service times have a negative exponential distribution. - How many channels are there?
- Do the customers come from a finite source or from an infinite source?
- Which queue model is appropriate here?
- What is the value of
![](/sites/dl/free/0072443901/24520/Image306.gif) Image306 (0.0K)Image306 ? - What is the interpretation of
![](/sites/dl/free/0072443901/24520/Image306.gif) Image306 (0.0K)Image306 ? - What is the value of
![](/sites/dl/free/0072443901/24520/Image307.gif) Image307 (0.0K)Image307 ? - What is the interpretation of
![](/sites/dl/free/0072443901/24520/Image307.gif) Image307 (0.0K)Image307 ? - What is the system utilization,
![](/sites/dl/free/0072443901/24520/Image308.gif) Image308 (0.0K)Image308 ?
Is this a feasible system? - What is the proportion of idle time?
- What is the mean number of customers being served?
- What is the expected number of customers waiting for service?
- What is the expected duration of the wait?
- What is the mean number of customers in the system?
- What is the mean time that a customer spends in the system?
- What is the probability that the system will be idle?
- What is the probability that there will be one car in the system?
- What is the probability that there will be three cars in the system?
2. In a factory, the parts to be copper-plated must be immersed in an electrolytic solution so that copper ions will be deposited on the part. The company has one electrolytic bath, in which each part is submerged for 20 minutes. Parts to be copper-plated arrive at a mean rate of 20 per 8-hour shift, according to a Poisson distribution. - How many channels are there?
- Do the parts come from a finite source or from an infinite source?
- Which queue model is appropriate here?
- What is the value of
![](/sites/dl/free/0072443901/24520/Image306.gif) Image306 (0.0K)Image306 ? - What is the interpretation of
![](/sites/dl/free/0072443901/24520/Image306.gif) Image306 (0.0K)Image306 ? - What is the value of
![](/sites/dl/free/0072443901/24520/Image307.gif) Image307 (0.0K)Image307 ? - What is the interpretation of
![](/sites/dl/free/0072443901/24520/Image307.gif) Image307 (0.0K)Image307 ? - What is the system utilization,
![](/sites/dl/free/0072443901/24520/Image309.gif) Image309 (0.0K)Image309 ?
Is this a feasible system? - What is the proportion of idle time?
- What is the expected number of parts waiting for copper-plating?
- What is the expected duration of the wait?
- What is the mean number of parts in the system?
- What is the mean time that a part spends in the system?
3. A bank has an array of five drive-in windows, some or all of which may be open from 10 a.m. to 3 p.m. on normal business days. At each window, the teller requires a mean of 2.4 minutes to serve each customer; service times have a negative exponential distribution. Customers arrive at a mean rate of 65 per hour, according to a Poisson distribution. (Use Table 19-4) - Do the customers come from a finite source or from an infinite source?
- What is the value of
![](/sites/dl/free/0072443901/24520/Image306.gif) Image306 (0.0K)Image306 ? - What is the interpretation of
![](/sites/dl/free/0072443901/24520/Image306.gif) Image306 (0.0K)Image306 ? - What is the value of
![](/sites/dl/free/0072443901/24520/Image307.gif) Image307 (0.0K)Image307 ? - What is the interpretation of
![](/sites/dl/free/0072443901/24520/Image307.gif) Image307 (0.0K)Image307 ? - What is the minimum number of drive-in windows which should be open to have
an underloaded system?
- What is the expected number of customers waiting for service with three drive-in
windows open?
- What is the expected duration of the wait?
- What is the expected duration of the wait if all three open windows are busy.?
- What is the probability that the system will be idle?
- What is the probability that an arrival will have to wait for service?
4. Rework Problem 3 with five drive-in windows open. - What is the expected number of customers waiting for service?
- What is the expected duration of the wait?
- What is the expected duration of the wait if all five windows are busy?
- What is the probability that the system will be idle?
- hat is the probability that an arrival will have to wait for service?
5. The bus company in a small city has a fleet of 10 buses and a single mechanic to service the buses as needed. Buses break down or require periodic maintenance at a mean rate of one every other day, seven days a week. The mechanic works five days per week (40 hours), and the mean time required to service one bus is four hours, with a negative exponential distribution. Buses which break down on Saturday or Sunday are parked until the mechanic can attend to them in the following week. - How many channels are there?
- Should the bus arrivals be treated as coming from a finite source or from
an infinite source?
- Which waiting line model is appropriate here?
- What is the number of potential customers, N?
- What is the value of the mean service time, T?
- What is the value of the mean time between service calls, U?
- What is the value of the service factor, X?
- What is the value of the efficiency factor, F?
- What is the expected number of buses running, J?
- What is the mean number of buses being serviced, H?
- What is the mean number of buses waiting for the mechanic, L?
1. Check to see that N = J + L + H. 6. A typing pool for a large corporation consists of six typists, each of whom can type at the rate of 12 pages per hour, on the average, with a Poisson distribution. The pool receives various kinds of typing jobs from a very large number of executives and managers. The jobs are classified into four priority levels as they arrive. Each level is shown below, with the mean of its particular arrival rate. Typing jobs arrive according to Poisson distributions: first priority: correspondence with customers (to be typed first); ![](/sites/dl/free/0072443901/24520/Image310.gif) Image310 (0.0K)Image310
pages per hr. second priority: correspondence with suppliers; ![](/sites/dl/free/0072443901/24520/Image311.gif) Image311 (0.0K)Image311
pages per hr. third priority: correspondence with federal, state, and local governments
(other than that dealing with sales);![](/sites/dl/free/0072443901/24520/Image312.gif) Image312 (0.0K)Image312
pages per hr. fourth priority: internal memoranda and communications and miscellaneous
correspondence; ![](/sites/dl/free/0072443901/24520/Image313.gif) Image313 (0.0K)Image313 pages per hr. - Which set of waiting line formulas should be used here?
- What is s?
- What is the value of
![](/sites/dl/free/0072443901/24520/Image314.gif) Image314 (0.0K)Image314 ? - What is
![](/sites/dl/free/0072443901/24520/Image315.gif) Image315 (0.0K)Image315 ? - What is the value of
![](/sites/dl/free/0072443901/24520/Image307.gif) Image307 (0.0K)Image307 ? - What is
![](/sites/dl/free/0072443901/24520/Image316.gif) Image316 (0.0K)Image316 ? - What is the system utilization,
![](/sites/dl/free/0072443901/24520/Image317.gif) Image317 (0.0K)Image317 ? - On the average, how many pages are being typed at any one time?
- Determine the value of
![](/sites/dl/free/0072443901/24520/Image318.gif) Image318 (0.0K)Image318 ? - Determine the value of A
- Determine these values:
![](/sites/dl/free/0072443901/24520/Image319.gif) Image319 (0.0K)Image319
![](/sites/dl/free/0072443901/24520/Image320.gif) Image320 (0.0K)Image320
![](/sites/dl/free/0072443901/24520/Image321.gif) Image321 (0.0K)Image321
![](/sites/dl/free/0072443901/24520/Image322.gif) Image322 (0.0K)Image322
![](/sites/dl/free/0072443901/24520/Image323.gif) Image323 (0.0K)Image323 - Determine these values:
![](/sites/dl/free/0072443901/24520/Image324.gif) Image324 (0.0K)Image324
![](/sites/dl/free/0072443901/24520/Image325.gif) Image325 (0.0K)Image325
![](/sites/dl/free/0072443901/24520/Image326.gif) Image326 (0.0K)Image326
![](/sites/dl/free/0072443901/24520/Image327.gif) Image327 (0.0K)Image327 - Find the average time in the system for pages in each priority class.
- Find the average number of pages in the queue for each priority class.
- Discuss the results of the above analysis.
7. Refer to problem #1. - What line length will not be exceeded 95% of the time?
- Refer to problem #3. What line length will not be exceeded 98% of the time?
Solutions 1. a. M = 1 channel.
b. An infinite source.
c. Single channel, single phase, Poisson arrivals, negative exponential service
times.
d. ![](/sites/dl/free/0072443901/24520/Image328.gif) Image328 (0.0K)Image328
cars per hour.
e. ![](/sites/dl/free/0072443901/24520/Image314.gif) Image314 (0.0K)Image314
is the mean number of customers who demand service every hour.
f. ![](/sites/dl/free/0072443901/24520/Image329.gif) Image329 (0.0K)Image329
minutes and
cars per hour.
g. ![](/sites/dl/free/0072443901/24520/Image307.gif) Image307 (0.0K)Image307
is the mean number of cars which the teller can handle in one hour.
h. ![](/sites/dl/free/0072443901/24520/Image331.gif) Image331 (1.0K)Image331 ,
which is less than 1. The system is feasible.
i. The proportion of idle time = ![](/sites/dl/free/0072443901/24520/Image332.gif) Image332 (1.0K)Image332
j. ![](/sites/dl/free/0072443901/24520/Image333.gif) Image333 (1.0K)Image333
customers.
k. ![](/sites/dl/free/0072443901/24520/Image334.gif) Image334 (1.0K)Image334
cars in the queue.
l. ![](/sites/dl/free/0072443901/24520/Image335.gif) Image335 (1.0K)Image335
hrs. or 2.20 min.
m. ![](/sites/dl/free/0072443901/24520/Image336.gif) Image336 (1.0K)Image336
cars.
n. ![](/sites/dl/free/0072443901/24520/Image337.gif) Image337 (1.0K)Image337
min.
o. p(0 cars) ![](/sites/dl/free/0072443901/24520/Image338.gif) Image338 (1.0K)Image338 .
p. p(1 car) ![](/sites/dl/free/0072443901/24520/Image339.gif) Image339 (1.0K)Image339 .
q. p(3 cars) ![](/sites/dl/free/0072443901/24520/Image340.gif) Image340 (1.0K)Image340
.
2. a. M = 1 channel.
b. An infinite population.
c. Single channel, single phase, Poisson arrivals, constant service times.
d. ![](/sites/dl/free/0072443901/24520/Image341.gif) Image341 (0.0K)Image341
parts per shift or 2 per hour.
e. ![](/sites/dl/free/0072443901/24520/Image342.gif) Image342 (0.0K)Image342
is the mean number of parts which arrive per time period.
f. ![](/sites/dl/free/0072443901/24520/Image343.gif) Image343 (0.0K)Image343
min. and ![](/sites/dl/free/0072443901/24520/Image344.gif) Image344 (1.0K)Image344
parts per shift.
g. ![](/sites/dl/free/0072443901/24520/Image307.gif) Image307 (0.0K)Image307 is
the maximum number of pieces which can be electro-plated per shift. (Compare
this with 1-g.)
h. ![](/sites/dl/free/0072443901/24520/Image345.gif) Image345 (1.0K)Image345 .
The electrolytic bath is in use 67% of the time and system is feasible.
i. The proportion of idle time ![](/sites/dl/free/0072443901/24520/Image346.gif) Image346 (1.0K)Image346
j. ![](/sites/dl/free/0072443901/24520/Image347.gif) Image347 (1.0K)Image347 .
k. ![](/sites/dl/free/0072443901/24520/Image348.gif) Image348 (1.0K)Image348 parts.
l. ![](/sites/dl/free/0072443901/24520/Image349.gif) Image349 (1.0K)Image349 .of
an 8-hour shift, or .0417(8) = .3333 hrs. or 20 minutes.
m. ![](/sites/dl/free/0072443901/24520/Image350.gif) Image350 (1.0K)Image350
parts.
n. ![](/sites/dl/free/0072443901/24520/Image351.gif) Image351 (1.0K)Image351
min. 3. a. An infinite source.
b. ![](/sites/dl/free/0072443901/24520/Image352.gif) Image352 (0.0K)Image352
customers per hr.
c. ![](/sites/dl/free/0072443901/24520/Image314.gif) Image314 (0.0K)Image314
is the mean number of customers who demand service at the array of drive-in
windows per hour.
d. At each window, ![](/sites/dl/free/0072443901/24520/Image353.gif) Image353 (0.0K)Image353
min. and ![](/sites/dl/free/0072443901/24520/Image354.gif) Image354 (0.0K)Image354 customers
per hr.
e. ![](/sites/dl/free/0072443901/24520/Image307.gif) Image307 (0.0K)Image307
is the mean number of customers which any one teller can handle in one hour.
f. ![](/sites/dl/free/0072443901/24520/Image309.gif) Image309 (0.0K)Image309
must be less than 1; hence ![](/sites/dl/free/0072443901/24520/Image355.gif) Image355 (1.0K)Image355
and M must be at least 3.
g. Use Table 19-4 in your textbook. ![](/sites/dl/free/0072443901/24520/Image356.gif) Image356 (1.0K)Image356
and ![](/sites/dl/free/0072443901/24520/Image357.gif) Image357 (0.0K)Image357
cars.
h. ![](/sites/dl/free/0072443901/24520/Image358.gif) Image358 (1.0K)Image358
hrs. = 4.55 min.
i. ![](/sites/dl/free/0072443901/24520/Image359.gif) Image359 (1.0K)Image359
hrs. = 6 min.
j. Use table 19-4; p(0) = .035.
k.
l. ![](/sites/dl/free/0072443901/24520/Image360.gif) Image360 (1.0K)Image360 4. a. Use Table 19-4; ![](/sites/dl/free/0072443901/24520/Image361.gif) Image361 (1.0K)Image361
and ![](/sites/dl/free/0072443901/24520/Image362.gif) Image362 (0.0K)Image362
cars.
b. ![](/sites/dl/free/0072443901/24520/Image363.gif) Image363 (1.0K)Image363
hrs. = 8.9 sec.
c. ![](/sites/dl/free/0072443901/24520/Image364.gif) Image364 (1.0K)Image364
hrs. = 60.1 sec.
d. Use Table 19-4; p(0) = .072.
e. ![](/sites/dl/free/0072443901/24520/Image365.gif) Image365 (1.0K)Image365
.
5. a. M = 1 channel, namely the mechanic.
b. The customers come from the finite fleet of 10 buses.
c. The finite source model.
d. N = 10 buses.
e. T = 4 hrs.
f. Presumably the buses run 24 hours a day, 7 days a week. U = 168 hrs./3.5
breakdowns = 48 hrs. between breakdowns.
g. X = T/(T + U) = 4/(4 + 48) = .077.
h. Use table 19-7 in your textbook. For X = .077 and M = 1, F » .973.
i. J = N(F)(1 - X) = 10(.973)(1 - .077) = 8.98 buses.
j. H = F(N)(X) = .973(10)(.077) = .75 buses.
k. L = N(1 - F) = 10(1 - .973) = .27 buses.
l. Check: N = J + L + K = 8.98 + .75 + .27 = 10 buses.![](/sites/dl/free/0072443901/24520/Image366.gif) Image366 (0.0K)Image366 6. a. The multiple-channel, multiple-priority model.
b. s = 6 typists.
c. ![](/sites/dl/free/0072443901/24520/Image367.gif) Image367 (1.0K)Image367
pages per hr.
d. ![](/sites/dl/free/0072443901/24520/Image368.gif) Image368 (1.0K)Image368
hrs =66.67 sec.
e. ![](/sites/dl/free/0072443901/24520/Image369.gif) Image369 (0.0K)Image369
pages per hr.
f. ![](/sites/dl/free/0072443901/24520/Image370.gif) Image370 (0.0K)Image370
hrs. = 5 min.
g. ![](/sites/dl/free/0072443901/24520/Image371.gif) Image371 (1.0K)Image371 .
h. ![](/sites/dl/free/0072443901/24520/Image372.gif) Image372 (1.0K)Image372
pages.
i. ![](/sites/dl/free/0072443901/24520/Image373.gif) Image373 (0.0K)Image373
customers (from Table 19-4).
j. ![](/sites/dl/free/0072443901/24520/Image374.gif) Image374 (1.0K)Image374 .
k. ![](/sites/dl/free/0072443901/24520/Image375.gif) Image375 (0.0K)Image375
![](/sites/dl/free/0072443901/24520/Image376.gif) Image376 (1.0K)Image376
![](/sites/dl/free/0072443901/24520/Image377.gif) Image377 (1.0K)Image377
![](/sites/dl/free/0072443901/24520/Image378.gif) Image378 (1.0K)Image378 .
![](/sites/dl/free/0072443901/24520/Image379.gif) Image379 (1.0K)Image379
l. ![](/sites/dl/free/0072443901/24520/Image380.gif) Image380 (1.0K)Image380
hrs. = 24 sec.
![](/sites/dl/free/0072443901/24520/Image381.gif) Image381 (1.0K)Image381 hrs. = 32 sec.
![](/sites/dl/free/0072443901/24520/Image382.gif) Image382 (1.0K)Image382 hrs. = 40 sec.
![](/sites/dl/free/0072443901/24520/Image383.gif) Image383 (1.0K)Image383 hrs. = 2.1 min.
m. Priority
n. Priority o. This system looks feasible; ![](/sites/dl/free/0072443901/24520/Image392.gif) Image392 (0.0K)Image392
and the jobs in each priority class are typed within a reasonable time, on the
average.
7. a. ![](/sites/dl/free/0072443901/24520/Image393.gif) Image393 (1.0K)Image393
ln K = ln 0.2185 = -1.52
ln
= ln 0.48 = -0.73
number of customers = ![](/sites/dl/free/0072443901/24520/Image395.gif) Image395 (1.0K)Image395 b. ![](/sites/dl/free/0072443901/24520/Image396.gif) Image396 (1.0K)Image396
ln K = ln 0.030 = -3.506
ln ![](/sites/dl/free/0072443901/24520/Image394.gif) Image394 (0.0K)Image394
= ln 0.867 = -0.143
number of customers = n = ln K/ln ![](/sites/dl/free/0072443901/24520/Image394.gif) Image394 (0.0K)Image394 =
-3.506/-0.143 = 24.52 ![](/sites/dl/free/0072443901/24520/Image120.gif) Image120 (0.0K)Image120
25.00. |