Consider the Arnold’s Muffler Shop example in Figure 9.8. Arnold has three chairs in the waiting room area.

(a) If all customers who bring their cars into the shop want to wait in the waiting room, what is the probability that someone will not have a seat?

(b) Arnold wants to provide a comfortable waiting room experience, so he would prefer that the probability that a customer has to stand is no more than 3%. How many more chairs will he need to purchase? You may need to re-run the model in Excel to find out.

Figure 9.8

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M/M/s with s = 1 is the M/M/1 model. Problem title D E F 1 Arnold's Muffler Shop 2 Queuing Model 3 Both x and umust 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. MMis (Exponential Service Times) 4 6. 2 cars per hour 2 3 8 Input Data 9 Arrival rate () 10 Service rate () 11 Number of servers (s) Operating Characteristics Average server utilization (p) Average number of customers in the queue (L,) Average number of customers in the system (L) Average waiting time in the queue (W) Average time in the system (W) Probability (% of time) system is empty (P) Mechanic is busy 67% of the time. 0.6667 1.3333 2.0000 0.6667 1.0000 0.3333 12 One mechanic 13 0.67 hours 40 minutes 14 15 3 mufflers per hour 16 17 = 60 minutes 18 Probabilities Cumulative Probability Probability 0.3333 Number of Units 20 0.3333 0.2222 0.1481 21 1 0.5556 0.7037 0.8025 0.8683 0.9122 0.9990 0.9993 0.9995 22 2 23 3 0.0988 This is the probability that 0.0658 0.0439 24 4 there are s 4 cars in the 25 5. 36 16 0.0005 system. 37 38 17 18 0.0003 0.0002 39 19 0.0002 0.9997 40 20 0.0001 0.9998 122222