Subject: Strategic Decision with Management Sciences A restaurant’s reservation
ID: 364175 • Letter: S
Question
Subject: Strategic Decision with Management Sciences
A restaurant’s reservation agent takes reservations for dinner by telephone. If he is already on the phone when a patron calls to make a reservation, the incoming call is answered automatically, and the customer is asked to wait for the agent. As soon as the agent is free, the patron who has been on hold the longest is transferred to the agent to be served. Calls for reservations come in at a rate of about 15 per hour. The agent is able to make a reservation in an average of three minutes. (20 per hour) Calls tend to follow a Poisson distribution, and the times to make a reservation tend to be exponential. The agent is paid $15 per hour. The restaurant estimates that every minute a customer must wait to speak to the agent costs the restaurant $1.
1. How often is the reservation busy (his utilization rate)
2. What is the average number of callers waiting to make a reservation?
3. What is the average number of customers in the system?
4. What is the average time in minutes that diners must wait before their calls are transferred to the agent?
5. What is the average time in minutes for diners to complete their reservations?
6. What is the probability of a customer NOT having to wait for their call to be answered
7. What is the probability of there being 2 customers in the system?
8. What is the probability of there being 5 or more customers in the system?
9. What is the purpose of studying queuing problems in Management Science?
In Arial rate ( Service rate () ata rat haracteristics Average server utilization (P Average number of customers in the queue (La) Average number of customers in the system ( Average waiting time in the queue (We) Average time in the system Probabili 0.7500 22500 3.0000 0.1500 0.2000 0.2500 of time) system is e Cost per hour Cw XVVa Csxs Probabilities Cumulative Number ofUnits Probability Probability 0.2500 0.4375 0.5781 0.6836 0.2500 0.1875 0.1406 0.1055 0.0791 0.0593 0.0445 0.0334 0.0250 0.8220 0.8665 0.8999 0.9249Explanation / Answer
Arrival rate, = 15
Service, = 20
1). utilization rate, = / = 15/20 = 0.75
2). Average number of callers waiting (Lq) = 2/(*(-)) = 152/(20*(20-15)) = 2.25
3). Average number of customers in the system (L) = /(-) = 15/(20-15) = 3
4). Average time in minutes that diners must wait before their calls are transferred to the agent (Wq) = Lq/ = 2.25/15 = 0.15 hour or 9 minutes
5). Average time in minutes for diners to complete their reservations (W) = L/ = 3/15 = 0.2 hour or 12 minutes
6). Probability of a customer NOT having to wait for their call to be answered (P0) = 1-/ = 1-0.75 = 0.25
7). What is the probability of there being 2 customers in the system (P2) = (1-)*2 = (1-0.75)*0.752 = 14.06 %
8). What is the probability of there being 5 or more customers in the system = 1 - (P0+P1+P2+P3+P4)
= 1 - ((1-0.75)*0.750 + (1-0.75)*0.751 + (1-0.75)*0.752 + (1-0.75)*0.753 + (1-0.75)*0.754 )
= 0.2373 or 23.73 %
9). Purpose of studying queuing problems in management science is to analyse and find tradeoffs between cost of employing resources and the cost of waiting in order to make economically justified decisions.