Question

A government agency counselling customers (citizens) subjected to domestic violence employs four lawyers. The agency manages complaints from three types of customers. The time a lawyer spends with each type of customers is exponentially distributed, with a mean of 15 minutes. Inter-arrival times for each customer type are exponential, with the average number of arrivals per hour for each customer type being observed as follows: type 1, 3 customers per hour; type 2, 5 customers per hour; and type 3, 3 customers per hour. Assume that type 1 customers have the highest priority, and type 3 customers have the lowest priority. Preemption is not allowed. a) What is the average length of time that each type of customers must wait before seeing a lawyer?

Answer 1

a.

λ = arrival rate

avg. time per customer = 15 min = 0.25 hour

µ = mean service rate = no. of lawyer * 1/(avg. time per customer) = 4*1/(0.25) = 16

average waiting time beforre they get to see lawyer

= average time spent waiting in line

= pW

= (λ/µ)*1/(µ-λ)

for type 1 :

λ = 3 as they have to wait only for other type 1 customers

average waiting time before they get to see lawyer for type 1

= (λ/µ)*1/(µ-λ)

= (3/16)*1/(16-3)

= 0.0144

average wait time for type 1 customer = 0.0144 hour

for type 2 :

λ = 3+5 as they have to wait only for type 1 and other type 2 customers

average waiting time before they get to see lawyer for type 2

= (λ/µ)*1/(µ-λ)

= (8/16)*1/(16-8)

= 0.0625

average wait time for type 2 customer = 0.0625 hour

for type 3 :

λ = 3+5+3 as they have to wait for type 1, type 2 and other type 3 customers

average waiting time before they get to see lawyer for type 3

= (λ/µ)*1/(µ-λ)

= (11/16)*1/(16-11)

= 0.1375

average wait time for type 3 customer = 0.1375 hour

(please UPVOTE for my hardwork)