1) Consultancy firm

A consultancy firm has been experiencing a very stable workload over the last few years. On average, the firm accepts 60 engagements every year and has 13 active engagements at any point in time.

1) What is the average duration of an engagement for the consultancy firm (Please explain your calculation) months

2) Suppose the management believe that the scope (i.e. length) of the projects they are undertaking is appropriate. However, in an attempt to grow the business they wish to accept 80 engagements per year. In that case, what would be the average number of active engagements at any time? (Please explain your calculation) Projects

2) (Capacity and Queues) Supermarket

A large supermarket experiences stable demand on Saturday afternoons. During this period the store has 4 staffed registers and 5 self-checkout machines located in two separate check-out areas. Each register requires an average of 4 minutes to serve a customer and self-checkout takes 5.5 minutes on average. However, this activity time is subject to unpredictable variability. Specifically, both checkout types have a coefficient of variation of 0.75.

The arrival rate to the self-checkout area is 1 customer every 2 minutes with coefficient of variation equal to 1. The arrival rate to the staffed registers is 0.6 customers per minute, also with a coefficient of variation equal to 1.

Customers entering the self-checkout area stand in a single queue operating on a first-in-first-out basis for any of the 5 self-checkout machines (i.e., a single pooled queue).

a) What is the utilization for the two check-out types? Please explain your calculation

Utilization of self-checkout:

Utilization of registers:

b) What is the average number of customers waiting in the self-checkout area? Please explain your calculation

Customers

c) What is the average waiting time in the self-checkout area? Please explain your calculation

Minutes:

d) What is the average customer waiting time for a staffed register? Please explain your calculation

Minutes:

e) If there were 4 separate queues for each one of the staffed registers, each seeing 9 customers each hour on average, with a coefficient of variation equal to 1 (i.e. separate dedicated waiting lines), what would be the average waiting time per customer in each queue? Please explain your calculation Minutes:

3) Queue Buildup) Flight 77 check-in counter

Flight 77 has just opened for check-in. There is a dedicated check-in area for this flight, consisting of several check-in counters pulling passengers from a single queue operating on a first-in-first-out basis. The check-in counters have a total processing capacity of 80 passengers per hour. Although the airline recommends that passengers arrive between 2h and 1h30min before departure, in practice passengers begin arriving 3 hours before departure, and the last passenger arrives at the queue about 1 hour before the plane departs. During the 30-minute recommended arrival period, passengers arrive at a rate of 140 passengers per hour. During the rest of the arrival period between 3h and 1h before the flight departs (that is excluding the 30 minute recommended arrival period), passengers arrive at a rate of 50 passengers per hour. For simplicity assume that there is a single class in this flight, and that all passengers need to go through the same check-in process.

Ignoring any unpredictable variability in both arrivals and processing times and assuming a continuous/fluid and deterministic flow of passengers, construct a build-up diagram to answer the following questions:

a) What is the largest number of passengers waiting in the queue at any point during the check-in process? Please explain your calculation

Passengers:

b) How many passengers are still in the queue 1 hour before the flight departs? Please explain your calculation

Passengers:

c) How much time before the plane's departure will the very last passenger check-in? Please explain your calculation Minutes before departure: