Sojourn Time Distributions
The term product form has also been used to refer to the sojourn time distribution in a cyclic queueing system, where the time spent by jobs at M nodes is given as the product of time spent at each node. In 1957 Reich showed the result for two M/M/1 queues in tandem, later extending this to n M/M/1 queues in tandem and it has been shown to apply to overtake–free paths in Jackson networks. Walrand and Varaiya suggest that non-overtaking (where customers cannot overtake other customers by taking a different route through the network) may be a necessary condition for the result to hold. Mitrani offers exact solutions to some simple networks with overtaking, showing that none of these exhibit product-form sojourn time distributions.
For closed networks, Chow showed a result to hold for two service nodes, which was later generalised to a cycle of queues and to overtake–free paths in Gordon–Newell networks.
Read more about this topic: Product Form Solution
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