## 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 =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.