Q1(Ref Q.5-44 p221-222: Agents at the security gate at the Hurtsfield County Reg
ID: 326075 • Letter: Q
Question
Q1(Ref Q.5-44 p221-222: Agents at the security gate at the Hurtsfield County Regional Airport are able to check passengers and process them through the security gate at the rate of 52 per hour (Poisson distributed) Passengers arrive at the gate throughout the day at the rate of 45 per hour (Poisson distributed) (a) Determine the following performance measures: i Average utilization of the gate . Average number of passengers waiting in the systenm fi Average time a passenger spends in the system Iv Probability that there are no passengers in the system v. Probability of 3 customers in the system vi Probability of 5 customers in the system viL Probability that an arrival has to wait for service (b) The passenger traffic arriving at the airport security gate varies significantly during the day and flight takeoffs tend to cluster making the passenger traffic very heavy at specific times while at other times there is ittle or no passenger traffic through the security gate. At the times leading up to flight takeoffs passengers arrive at a rate of 125 per hour (Poisson distributed). The airport authority is considering the option of in order to accommodate this increased level of passenger traffic. Design a waiting line system that can accommodate this increased level of passenger traffic. Compare the following performance measures of this new system with the current one: i Average utilization of the gate ii. Expected number of passengers waiting in the system i. Expected time a passenger spends in the system iv. Probability that there are no passengers in the system v. Probability of 3 customers in the system vi Probability of 5 customers in the system vi Probability that an arrival has to wait for service (c) The manager has decided to see the financial implications of decreasing/increasing the number of security gate on the performance of the system? If the cost of hiring an additional security agent is $15/hour and the cost of keeping a passenger in waiting line is $0.50/min, determine the optimal number of agents that the manager should use? Show all the relevant calculations.Explanation / Answer
1.Determine the average waiting time and the waiting line.
? = mean number of arrivals per time period = 45/hr ;
µ = mean number of people served per time period = 52/hr
Ls = average number of passengers in the system (waiting and being served) = (?/µ- ?) = 45/7 = 6.43 or 6
Lq = average number of passengers in the queue= (?/µ) (?/µ- ?) = (45/52) × 6.43 = 5.56 or 6
Ws = average time a passenger spends in the system (waiting time plus service time)= 1/(µ- ?) = 1/7= 0.1429 hr or 8.57 mins
Wq = average time a passenger spends waiting in the queue= (?/µ)(1/µ- ?) = (45/52) × 0.143 = 0.124hr or 7.42mins
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