Question

Trucks are required to pass through a weighing station so that they can be checked for weight violations. Trucks arrive at the station at the rate of 18 an hour between 7:00 p.m. and 9:00 p.m. Currently two inspectors are on duty during those hours, each of whom can inspect 15 trucks an hour.

a. How many trucks would you expect to see at the weighing station, including those being inspected? Ls=_____trucks

b.If a truck was just arriving at the station, how long could the driver expect to be at the station, including time waiting in line and being weighed? Ws____hrs

c. What is the probability that both inspectors would be busy at the same time? Pw=

d. How long, on average, would a truck that is not immediately inspected have to wait? Wa=_____hrs

Answer 1

This is M/M/s queue system with following parameters,

Arrival rate, = 18 trucks per hour

Service rate, = 15 trucks per hour

Number of servers (inspectors), M = 2

/ = 18/15 = 1.2

Refer following table for / = 1.2 and M = 2

Lq = 0.675

a) Expected number of trucks at the weighing station, Ls = Lq + / = 0.675+1.2 = 1.875 trucks

b) Expected time at the weighing station, Ws = L/ = 1.875/18 = 0.1042 hour

c) Probability that both inspectors would be busy at the same time, Pw = /(M) = 18/(2*15) = 0.60

d) Average waiting time, Wq = Lq/ = 0.675/18 = 0.0375 hour