Question

An average of 40 cars per hour (interarrival times are exponentially distributed) are tempted to use the drive-in window at the Hot Dog King restaurant. If a total of more than 4 cars are in line (including the car at the window) a car will not enter the line. It takes an average of 4 minutes (exponentially distributed) to serve a car.

1. (a) What is the average number of cars waiting for the drive-in window (not including a car at the window)?

2. (b) On the average, how many cars will be served per hour?

3. (c) I have just joined the line at the drive-in window. On the average, how long will it be

   before I have received my food?

Answer 1

Salution: Given that Неле, Аллia. P= 40 caus | Fa51 . Service rate u= _ coll minute = 15 Carol Lower Also, the traffic intensity CP) Cs pa ū - 40 = 2.67 4 . The steady state probability is defined below To - IpCHU Tij = Pito (j = 1, 2, 3, 4) TJ = 0 () = 5,6) al. Lq -L-LS- 29 - average homber of cars waiting in line L - average number of car present in the quening - systen L = Denotes the average number of cars in service

- The average number of cars present in the quening systen Lis L = *[1-1+1) P + cat] Ci+pth) (1-P): Here, c=4; P=2.67. Thus subsituting values of c; p. L=2.67 [1-5 (2.674 + 4(2.6789 [!-(2.67}] C1-267). = 2.67 [1-5050:82) +4 (135169) (1-135.69) (1~2.67) 267X 289.67 (-134.69) (-1.67) 3. 438 The average number of cars in service is Ls = 1- To. Here. To denotes the steady state probability that o cars are present în service.

To = Low Tip : P in pcct) = 1-2-67 1-(2.675 = - 1.67 - 134.69 -0.012. Thus . L5 = 1-TO =1-0.012 = 0.987. Now, L-3438; 1s = 0.987. - so, . the average number of cars waiting in line is Lq = L-is = 3.348-0.987 > 2.45 b). Here Total capacity of system caq. Avival rate 9 = 40 carshowy. P = 2.67: TG = 0.012 The steady state probability that 4 cars are present in the system is

Tq = pt to =(2.67)* *0.012 = 0.61 so the average the number of cars served per hour is q (l- 7) = 40C1-61). = 15.6 car per hour .. c. W = L subsitute L=3:438;9-40,11 4 - 06), = 3.438 40(1-0.61) = 3.438 = 0.22 hours. 15.6 Thus on an average, it will take o.22 hour or 13.2 minute before I have recived my food. +