Problem 4 14.b

M/M/4 queue with arrival rate = 2.4 per minutes and service rate of = 0.7 per minute for each of the four individual servers.

Compute \(W_q\), \(Q\), \(L_q\), \(L\), and \(\rho\)

To calculate the numerical output of the M/M/4 system, I used the mmc utility:

Simio

For the simulated model in Simio, I used a single server with an Initial Capacity = 4.

Property Configutation of the Source Node

On the source node, the Interarrival time is set to Random Expontial with a mean value of 1/= 1/2.4

Property configuration of the server node

On the server node, the Initial Capacity is set to 4 and the processing time is set to Random Expontial with a mean value of 1/= 1/0.7

Output

I created an experiment where I ran 10 executions of the simulation. Simio model output below:

Metric Comparison

Metric Numerical Result Simulated Result
\(L\) : Avg Entities in System 7.6468 7.7044 (Avg NumberInSystem)
\(L_q\) : Avg Entities in Queue 4.2183 4.2698 (Avg NumberInStation)
\(W\) : Avg Time in System 3.1862 3.2072 (Avg TimeInSystem)
\(W_q\) : Avg Time in Queue 1.7576 1.7774 (Avg TimeInStation)
\(\rho\) : Utilization 0.8571 0.8586 (Avg ScheduledUtilization)