Submitted To Submitted By

Prof. Bibhuti Tripathi Gaurav Sharma (016)

a) Identify the customers and servers for this queuing system. How many servers does it currently have?

Customers: The customers in this queuing system are the TrucksServers: The servers are the crew members. There are 4 servers as there are 4 crew members currently.

b) Use the Excel template to find the various measures of performance for this queuing system with four members on the crew. (Set t = 1 hour in the Excel template for the waiting-time probabilities.)

Ws= Waiting time in the systemNo. of Serves= No. of crew memberWq=Waiting in the system

Given Information:Arrival rate(ƛ) 1Service Rate(µ) 4Members 4

No. of Servers Ws(hrs) Wq(hrs)

Waiting Cost in queue/hour($)

Loading/unloading Cost($)

Total Cost($)

4 0.333333333 0.083333 2.5 80 82.5

Member 4 3 2 Idle Time 0.25 0.333 0.5Arrival Rate P0 0.750000 0.666667 0.5

P1 0.187500 0.222222 0.25P2 0.046875 0.074074 0.125P3 0.011719 0.024691 0.0625P4 0.002930 0.008230 0.03125P5 0.000732 0.002743 0.015625P6 0.000183 0.000914 0.007813P7 0.000046 0.000305 0.003906P8 0.000011 0.000102 0.001953P9 0.000003 0.000034 0.000977P10 0.000001 0.000011 0.000488

(c) Repeat (b) with three members.(d) Repeat part (b) with two members.

No. of Servers Ws(hrs) Wq(hrs)

Waiting Cost in queue/hour ($)

Loading/unloading Cost($)

Total Cost($)

3 0.5 0.166667 5 60 652 1 0.5 15 40 551 #DIV/0! #DIV/0! #DIV/0! 20 #DIV/0!

(e) Should a one-member crew also be considered? ExplainIf the average arrival rate is equal to the service rate, the idle period can never be made up. Because the overall average arrival rate is equal to the service rate, eventually the trucks will arrive and there will be a huge pile up in the queue. Over time, this truck pile up will get longer and longer and the facility will eventually congest.

(f) Given the previous results, which crew size do you think management should choose?

The management should choose crew size of 2 as we can see that the total cost (waiting cost + Loading/unloading cost) is minimum in the case when the crew size is 2. The cost is 55$.

(g) Use the cost figures to determine which crew size would minimize the expected total cost per hour.

The management should choose crew size of 2 as we can see that the total cost (waiting cost + Loading/unloading cost) is minimum in the case when the crew size is 2. The cost is 55$.

(h) Assume now that the mean service rate of the crew is proportional to the square root of its size. What should the size be to minimize expected total cost per hour?

2nd Case(Mean Service Rate of crew proportional to the square root of Size) No. of Servers

Sqrt of Servers Ws(hrs) Wq(hrs)

Waiting Cost /hour

Loading/unloading Cost

Total Cost

4 2.0000 1.0000 0.5000 15.0000 80 95.00003 1.7321 1.3660 0.7887 23.6603 60 83.66032 1.4142 2.4142 1.7071 51.2132 40 91.21321 1.0000 #DIV/0! #DIV/0! #DIV/0! 20 #DIV/0!

The minimum expected total cost per hour will be achieved when the size of the crew is 2