Solution
| If you want the formulas and any calculations, select the corresponding cell and press F2(Function Key on key board), |
| It will show all calculations and formulas Automatically |
| Question: |
| 14-11 |
| The Rockwell Electronics Corporation retains a service crew to repair machine breakdowns that occur on an average of 𝜆=3 per day (approximately Poisson in nature), the crew can service an average of µ = 8 machines per day, with a repair time distribution that resembles the exponential distribution. |
| (a) What is the utilization rate of this service system? |
| (b) What is the average downtime for a machine that is broken? |
| (c) How many machines are waiting to be serviced at any given time? |
| (d) What is the probability that more than one machine is in the system? Probability that more than |
| two are broken and waiting to be repaired or being serviced? More than three? More than four? |
| Arrival rate (λ) |
3 |
Per day |
| Service rate (μ) |
8 |
Per day |
| (a) What is the utilization rate of this service system? |
| Solution: computation of the following |
| Utilization rate(U)=λ/μ |
| server utilization (U) |
37.50% |
| (b) What is the average downtime for a machine that is broken? |
| Solution: computation of the following |
| The average down time is the time that the machine waits to be serviced plus the time taken to repair the machine. |
| The average down time is given by W |
| W=1/1(μ-λ) |
| W |
0.2 |
Day |
| assuming 8 hrs/day |
1.6 |
Hours |
| (c) How many machines are waiting to be serviced at any given time? |
| Solution: computation of the following |
| Lq=λ^2/μ(μ-λ) |
| Lq |
0.225 |
Machines |
| (d) What is the probability that more than one machine is in the system? Probability that more than two are broken and waiting to be repaired or being serviced? More than three? More than four? |
| Solution: computation of the following |
| Pn>k=(λ/μ)^(k+1) |
| Pn>1 |
0.141 |
| Pn>2 |
0.053 |
| Pn>3 |
0.020 |
| Pn>4 |
0.007 |
Solution 2
| If you want the formulas and any calculations, select the corresponding cell and press F2(Function Key on key board), |
| It will show all calculations and formulas Automatically |
| Question: |
| From historical data, Harry’s car wash estimates that dirty cars arrive at the rate of 10 per hour all day Saturday. With a crew working the wash line, Harry figures that cars can be cleaned at the rate of one every five minutes. One car at a time is cleaned in this example of single-server waiting line. Assuming Poisson arrivals and exponential service times, find the |
| A) average number of cars in line |
| B) average time a car waits before it is washed |
| C) average time a car spends in the service system |
| D) utilization rate of the car wash |
| E) probability that no cars are in the system |
| Arrival rate |
10 |
cars Per hour |
| Service rate |
One car at every 5 minutes |
| Service rate |
12 |
cars per hour |
| Number of servers (s) |
1 |
| Entering above values in the Excelmodules Queuing models—->M/M/s, we get following results: |
| A) average number of cars in line |
| Avg no of cars in line(Lq) |
4.1666666667 |
| B) average time a car waits before it is washed |
| Avg waiting time in queue(Wq) |
0.4166666667 |
hours |
| Avg waiting time in queue(Wq) |
25 |
Mins |
| C) average time a car spends in the service system |
| Avg time in service system(W) |
0.5 |
hours |
| Avg time in service system(W) |
30 |
Mins |
| D) utilization rate of the car wash |
| Average utilization of service system |
0.8333333333 |
83.33 |
Percent |
| E) probability that no cars are in the system |
| Probability of no car in the system(P(0)) |
0.1666666667 |
Solution_Excel modules
| Harry’s car wash |
| Queuing Model |
M/M/s (Exponential Service Times) |
| Input Data |
|
|
Operating Characteristics |
| Arrival rate (l) |
10 |
|
Average server utilization (r) |
0.8333 |
| Service rate (m) |
12 |
|
Average number of customers in the queue (Lq) |
4.1667 |
| Number of servers (s) |
1 |
|
Average number of customers in the system (L) |
5.0000 |
|
|
|
Average waiting time in the queue (Wq) |
0.4167 |
|
|
|
Average time in the system (W) |
0.5000 |
|
|
|
Probability (% of time) system is empty (P0) |
0.1667 |
| 0 |
| Probabilities |
| Number of Units |
Probability |
Cumulative Probability |
| 0 |
0.1667 |
0.1667 |
| 1 |
0.1389 |
0.3056 |
| 2 |
0.1157 |
0.4213 |
| 3 |
0.0965 |
0.5177 |
| 4 |
0.0804 |
0.5981 |
| 5 |
0.0670 |
0.6651 |
| 6 |
0.0558 |
0.7209 |
| 7 |
0.0465 |
0.7674 |
| 8 |
0.0388 |
0.8062 |
| 9 |
0.0323 |
0.8385 |
| 10 |
0.0269 |
0.8654 |
| 11 |
0.0224 |
0.8878 |
| 12 |
0.0187 |
0.9065 |
| 13 |
0.0156 |
0.9221 |
| 14 |
0.0130 |
0.9351 |
| 15 |
0.0108 |
0.9459 |
| 16 |
0.0090 |
0.9549 |
| 17 |
0.0075 |
0.9624 |
| 18 |
0.0063 |
0.9687 |
| 19 |
0.0052 |
0.9739 |
| 20 |
0.0043 |
0.9783 |
| Computations |
| n or s |
(lam/mu)^n/n! |
Cumsum(n-1) |
term2 |
P0(s) |
Rho(s) |
Lq(s) |
L(s) |
Wq(s) |
W(S) |
| 0 |
1 |
| 1 |
0.8333333333 |
1 |
5 |
0.1666666667 |
0.8333333333 |
4.1666666667 |
5 |
0.4166666667 |
0.5 |
| 2 |
0.3472222222 |
1.8333333333 |
0.5952380952 |
0.4117647059 |
0.4166666667 |
0.175070028 |
1.0084033613 |
0.0175070028 |
0.1008403361 |
| 3 |
0.0964506173 |
2.1805555556 |
0.1335470085 |
0.432132964 |
0.2777777778 |
0.0221961787 |
0.855529512 |
0.0022196179 |
0.0855529512 |
| 4 |
0.0200938786 |
2.2770061728 |
0.0253817414 |
0.4343316753 |
0.2083333333 |
0.0029010774 |
0.8362344108 |
0.0002901077 |
0.0836234411 |
| 5 |
0.0033489798 |
2.2971000514 |
0.0040187757 |
0.434571213 |
0.1666666667 |
0.0003492888 |
0.8336826222 |
0.0000349289 |
0.0833682622 |
| 6 |
0.0004651361 |
2.3004490312 |
0.000540158 |
0.4345956968 |
0.1388888889 |
0.000037863 |
0.8333711963 |
0.0000037863 |
0.0833371196 |
| 7 |
0.0000553733 |
2.3009141673 |
0.0000628562 |
0.4345979946 |
0.119047619 |
0.0000036915 |
0.8333370248 |
0.0000003692 |
0.0833337025 |
| 8 |
0.0000057681 |
2.3009695406 |
0.0000064388 |
0.4345981918 |
0.1041666667 |
0.0000003254 |
0.8333336587 |
0.0000000325 |
0.0833333659 |
| 9 |
0.0000005341 |
2.3009753087 |
0.0000005886 |
0.4345982073 |
0.0925925926 |
0.0000000261 |
0.8333333594 |
0.0000000026 |
0.0833333359 |
| 10 |
0.0000000445 |
2.3009758428 |
0.0000000486 |
0.4345982084 |
0.0833333333 |
0.0000000019 |
0.8333333353 |
0.0000000002 |
0.0833333335 |
| 11 |
0.0000000034 |
2.3009758873 |
0.0000000036 |
0.4345982085 |
0.0757575758 |
0.0000000001 |
0.8333333335 |
0 |
0.0833333333 |
| 12 |
0.0000000002 |
2.3009758906 |
0.0000000003 |
0.4345982085 |
0.0694444444 |
0 |
0.8333333333 |
0 |
0.0833333333 |
| 13 |
0 |
2.3009758909 |
0 |
0.4345982085 |
0.0641025641 |
0 |
0.8333333333 |
0 |
0.0833333333 |
| 14 |
0 |
2.3009758909 |
0 |
0.4345982085 |
0.0595238095 |
0 |
0.8333333333 |
0 |
0.0833333333 |
| 15 |
0 |
2.3009758909 |
0 |
0.4345982085 |
0.0555555556 |
0 |
0.8333333333 |
1.34352048387924E-16 |
0.0833333333 |
| 16 |
0 |
2.3009758909 |
0 |
0.4345982085 |
0.0520833333 |
6.51218686419213E-17 |
0.8333333333 |
6.51218686419213E-18 |
0.0833333333 |
| 17 |
1.26721499130873E-16 |
2.3009758909 |
1.33253535168546E-16 |
0.4345982085 |
0.0490196078 |
2.98514163203532E-18 |
0.8333333333 |
2.98514163203532E-19 |
0.0833333333 |
| 18 |
5.86673607087373E-18 |
2.3009758909 |
6.15152908402294E-18 |
0.4345982085 |
0.0462962963 |
1.29778811625998E-19 |
0.8333333333 |
1.29778811625998E-20 |
0.0833333333 |
| 19 |
2.57312985564637E-19 |
2.3009758909 |
2.69116333526318E-19 |
0.4345982085 |
0.0438596491 |
5.36502185461152E-21 |
0.8333333333 |
5.36502185461152E-22 |
0.0833333333 |
| 20 |
1.07213743985266E-20 |
2.3009758909 |
1.1187521111506E-20 |
0.4345982085 |
0.0416666667 |
2.11394636204157E-22 |
0.8333333333 |
2.11394636204157E-23 |
0.0833333333 |
| 21 |
4.25451365020895E-22 |
2.3009758909 |
4.43031999939114E-22 |
0.4345982085 |
0.0396825397 |
7.95623609441516E-24 |
0.8333333333 |
7.95623609441516E-25 |
0.0833333333 |
| 22 |
1.61155820083672E-23 |
2.3009758909 |
1.67500537409801E-23 |
0.4345982085 |
0.0378787879 |
2.86596194812096E-25 |
0.8333333333 |
2.86596194812096E-26 |
0.0833333333 |
| 23 |
5.83897898853885E-25 |
2.3009758909 |
6.05848947682979E-25 |
0.4345982085 |
0.0362318841 |
9.89852884544816E-27 |
0.8333333333 |
9.89852884544816E-28 |
0.0833333333 |
| 24 |
2.02742325990932E-26 |
2.3009758909 |
2.10035215415067E-26 |
0.4345982085 |
0.0347222222 |
3.28348663103548E-28 |
0.8333333333 |
3.28348663103548E-29 |
0.0833333333 |
| 25 |
6.75807753303108E-28 |
2.3009758909 |
6.9911146893425E-28 |
0.4345982085 |
0.0333333333 |
1.04769859291578E-29 |
0.8333333333 |
1.04769859291578E-30 |
0.0833333333 |
| 26 |
2.16605049135612E-29 |
2.3009758909 |
2.23777401755996E-29 |
0.4345982085 |
0.0320512821 |
3.22030655322929E-31 |
0.8333333333 |
3.22030655322929E-32 |
0.0833333333 |
| 27 |
6.68534102270406E-31 |
2.3009758909 |
6.89824997247171E-31 |
0.4345982085 |
0.0308641975 |
9.54766585945925E-33 |
0.8333333333 |
9.54766585945925E-34 |
0.0833333333 |
| 28 |
1.98968482818573E-32 |
2.3009758909 |
2.05071810512395E-32 |
0.4345982085 |
0.0297619048 |
2.73386016760705E-34 |
0.8333333333 |
2.73386016760705E-35 |
0.0833333333 |
| 29 |
5.71748513846475E-34 |
2.3009758909 |
5.88664150350809E-34 |
0.4345982085 |
0.0287356322 |
7.56900547795275E-36 |
0.8333333333 |
7.56900547795275E-37 |
0.0833333333 |
| 30 |
1.58819031624021E-35 |
2.3009758909 |
1.6335671824185E-35 |
0.4345982085 |
0.0277777778 |
2.02841534558582E-37 |
0.8333333333 |
2.02841534558582E-38 |
0.0833333333 |
1. Both l and m must be RATES, and use the same time unit. For example, given a service time such as 10 minutes per customer, convert it to a service rate such as 6 per hour. 2. The total service rate (rate x servers) must be greater than the arrival rate.
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