## Assume service times are exponentially distributed with an average of

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:

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:

## During busy periods (9:00 a.m.-11:30 a.m. and 2:00 p.m.-3:45 p.m.),

During busy periods (9:00 a.m.-11:30 a.m. and 2:00 p.m.-3:45 p.m.), the central appointment office receives 40 calls per hour, on average. Each call takes an average of 3.11 minutes to serve. It is desired that at least 90 percent of calls are received without waiting. What is the minimum number of staff needed during these busy times?

When doctors referred their patients to the Lourdes Hospital in Binghamton, New York, for various services such as X-rays, their office had a tough time getting through to the centralized appointment office of Lourdes. Most of the time, the line was busy. The installation of a call waiting system did not improve the situation, because callers were put on hold for indefinite lengths of time. The poor service had resulted in numerous complaints.

One of the managers was put in charge of finding a solution, and a goal of answering at least 90 percent of calls without delay was set. The hospital was willing to employ more staff to receive calls. The manager studied this queueing problem by collecting data for 21 workdays during which additional staff was used to answer calls and no call received a busy signal or was put on hold. The number of calls per day ranged between 220 and 350, with no day-of-the-week seasonality. Most days, the number of calls was between 250 and 300. The average number of calls arriving during each 15-minute interval peaked at about 10 calls during the 9:00 a.m.-11:30 a.m. and 2:00 p.m.-3:45 p.m. periods. The 944 service duration’s during the data collection period had a distribution similar to exponential with a mean of 3.11 minutes. The manager also found out that previously the 6.5 full-time-equivalent employees usually spent half their time doing other tasks and turned off their phones while busy with other tasks. Using the multiple servers queueing model and a service goal of at least 90 percent probability of not having to wait, the manager determined the number of staff required during each 15-minute interval. When the original staffing levels were compared with the model-determined ones, it was discovered that more staff were required earlier in the day and later in the afternoon, and fewer were needed around noon. The problem was solved by rearranging work shifts.

## Many of a bank’s customers use its automated banking machine

Many of a bank’s customers use its automated banking machine (ABM). During the early evening hours in the summer months, customers arrive at the ABM at the rate of one every other minute (assume Poisson). Each customer spends an average of 90 seconds completing his or her transaction. Transaction times are exponentially distributed. Assume that the length of queue is not a constraint. Determine:

a. The average time customers spend at the machine, including waiting in line and completing transactions.

b. The probability that a customer will not have to wait upon arrival at the ABM.

c. Utilization of the ABM.

d. The probability that a customer waits four minutes or more in the line.

## With its cool alpine climate and majestic mountain ranges, featuring

With its cool alpine climate and majestic mountain ranges, featuring tremendous vertical drops and copious amounts of snow, Alberta is home to several world class ski resorts. At a small and highly exclusive resort located in the heart of Banff National Park, managers have been contending with customer complaints regarding excessive ski lift wait times during the winter peak season. An idea has been floated around to double the number of chairs on the lift and slow down its speed by half, which will still allow customers ample time to safely board and disembark. This means there will only be 100 feet between lift chairs, instead of 200 feet, and each trip to the top of the mountain will now take six minutes instead of three minutes. Management argues that customers will not mind having an additional three minutes to take in the breathtaking beauty of the Canadian Rockies. Using waiting line analysis, show that the proposed idea will help reduce the excessive waiting times. For simplicity, assume that there are currently four lift chairs, the average time to ski down is two minutes, and the same 10 skiers circulate continuously between waiting for the lift, being lifted, and skiing down the mountain.

## Contact a manager of a local business. Determine the average

Contact a manager of a local business. Determine the average arrival rate and average service rate at one of these operations: a fast-food restaurant, supermarket, post office, or bank branch.

a. For the arrival rate, observe for 15 one-minute intervals. Count the number of arrivals in each minute. Construct a frequency distribution for arrivals. Does it resemble a Poisson distribution? Calculate its mean.

b. Now determine the service durations. To do this, observe the service time of 15 customers (e.g., how many minutes a teller at a bank spent with each customer). Make a frequency distribution. Does it look like exponential distribution? Calculate its mean.

## The following is a list of activities/work packages andtheir precedence

The following is a list of activities/work packages and their precedence and duration, for a new-plant start-up project. Draw the precedence network, calculate earliest and latest times, and determine the project duration and critical activities.

# Solution:

Project duration = 17 months

Critical activities = B, E, H, K