Assume service times are exponentially distributed with an average of 1.1 minutes. How many booths should CBSA

Question:

Assume service times are exponentially distributed with an average of 1.1 minutes. How many booths should CBSA open during each hour of the day to assure average wait times in the queues are less than 15 minutes?

The Peace Arch border crossing between Vancouver and Bellingham, Washington, has two distinct facilities: one controlled by U.S. Customs and Border Protections officers for drivers heading south into the state of Washington, and the other controlled by Canadian Border Services Agency officers for drivers heading north into British Columbia. Note that commercial trucks are not allowed to use the Peace Arch crossing ( they should instead use the Pacific highway crossing). Consider the northbound vehicles. There are two types of cars that arrive at the crossing: Nexus holders and non-Nexus holders. Nexus holders are assigned booths 1 or 2 (when busy) and their processing time is very fast, typically less than half a minute each. The non-Nexus holders are assigned to seven other booths, depending on how busy it is, and their processing time is approximately 1.1 minutes each. Non-Nexus cars arrive from two lanes off Highway 15 and then line up to go into these booths.

We will focus on the non-Nexus queuing system. The total number of daily northbound non-Nexus car arrivals at the border crossing varies by the day of the week (it is busier on Fridays, Saturdays, and Sundays). The numbers of arrivals (volume) for May 2017 are displayed below:

Sunday Monday Tuesday Wednesday Thursday Friday Saturday 30 2 Volume Volume Volume Volume Volume Volume 2631.0 2328.0 2207.0 2592.0 3236.0 3748.0 8 7 9 10 11 12 13 Volume Volume 2591.0 Volume 2245.0 Volume Volume Volume Volume 3394.0 2232.0 2487.0 3635.0 3744.0 14 15 16 17 18 19 20 Volume

The pattern of arrivals is slightly different from day to day. The numbers of arrivals (volume) during each five-minute interval on May 30, 2017, are given below:

The number of arrivals during any hour of any day has a Poisson distribution and the average service time for non-Nexus cars varies only slightly based on the time of day (see http://www.sciencedirect.com/science/article/pii/S0965856416305080). The CBSA opens booths based on the time of day, the day of the week, and the expected total volume. There is some flexibility as the officers can be shifted between the booths and the secondary screening, which is performed in a nearby building. Consider the following problem. Suppose CBSA expects 3,500 non-Nexus cars on a specific day with the following percentage of this total during each hour: