Question

At a convenience store there is only one cashier. It is estimated that average service time by the cashier is 2 minutes, and the standard deviation of service time is 1 minute. The owner observed that on average customers arrive every 4 minutes, and the standard deviation of the time between two consecutive arrivals is also 4 minutes. Answer the following questions (or fill in the blanks):

a) E(A) = ________, E(S) = _________ (Note: please include the time unit)

b) σ(A) = ________ , σ(S) = _________ (Note: please include the time unit)

c) C_A = _________, C_S = ____________

d) Arrival rate λ=________________, Service rate μ=_____________ (Note: please include the unit)

e) The cashier’s utilization is ρ= ____________

f) Find the (long-run) average queue length and average customer waiting time.

g) suppose that the owner wants to reduce the average customer waiting time below 1 min. Should she increase the number of cashiers in the store? Suppose that the owner wants to reduce the average customer waiting time below 1 min. Should she increase the number of cashiers in the store?

Answer 1

a)

Average service time = 2 minutes

Standard deviation of service time = 1 minute

Average inter-arrival time = 4 minutes

Standard deviation of inter-arrival time = 4 minutes

a) E(A) = 4 minutes , E(S) = 2 minutes (Note: please include the time unit)

b) σ(A) = 4 minutes , σ(S) = 1 minute (Note: please include the time unit)

c) C_A = 4/4 = 1 , C_S = 1/2 = 0.5

d) Arrival rate λ= 1/4 = 0.25 per minute , Service rate μ= 1/2 = 0,5 per minute    (Note: please include the unit)

e) The cashier’s utilization is ρ = E(S) / E(A) = 2/4 = 0.5

f)

Average waiting time, Wq = E(S) * ρ/(1-ρ)*(C_A²+C_S²)/2

= 2*0.5/(1-0.5)*(1²+0.5²)/2

= 1.25 minutes

Average queue length, Lq = Wq/E(A)

= 1.25/4

= 0.31