How to use Tandem Queueing Network Process?

Question

Mathematica provides QueueingNetworkProcess and QueueingProcess. However, I can't seem to figure out how to create a tandem queuing process: whenever a customer arrives, he first needs to go to server 1, then to server 2, then to server 3.

any help is greatly appreciated!

Answer 1

1) Using QueueingNetworkProcess to define such a queuing process

With the customer arrival rate arrivalRate and the service rates at each server being serviceRate1, serviceRate2, and serviceRate3, respectively, you can define this queuing process by

g = {arrivalRate, 0, 0}; 
m = {serviceRate1, serviceRate2, serviceRate3};
r = {{0, 1, 0}, {0, 0, 1}, {0, 0, 0}};
c = {1, 1, 1};

queuingProcess = QueueingNetworkProcess[g, r, m, c];

If you then want to compute the probability of e.g. having one customer at each server you can use:

sDist = StationaryDistribution[queuingProcess];
PDF[sDist, {1, 1, 1}]

To get the performance measures at all three nodes for some numerical values one can use:

Table[QueueProperties[{queuingProcess /. {arrivalRate -> 5., serviceRate1 -> 7., 
  serviceRate2 -> 6.5, serviceRate3 -> 8.3}, i}], {i, 3}]

If you want to perform calculations for several different rates, starting with a function like

queuingProcessF[arrivalRate_, {serviceRate1_, serviceRate2_, serviceRate3_}] := 
 QueueingNetworkProcess[{arrivalRate, 0, 0}, {{0, 1, 0}, {0, 0, 1}, {0, 0, 0}}, 
  {serviceRate1, serviceRate2, serviceRate3}, {1, 1, 1}]

might be more appropriate, though.

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2) Using the product of the individual QueueingProcesses

You can define the same queuing process by first defining each individual queue

q1 = QueueingProcess[arrivalRate, serviceRate1];
q2 = QueueingProcess[arrivalRate, serviceRate2];
q3 = QueueingProcess[arrivalRate, serviceRate3];

and then use ProductDistribution to generate the same StationaryDistribution:

sDist2 = ProductDistribution[StationaryDistribution[q1], StationaryDistribution[q2], 
           StationaryDistribution[q3]];

PDF[sDist2, {1, 1, 1}] == PDF[sDist, {1, 1, 1}]

True