Bottlenecks in Markovian Queueing Networks
Department of Mathematics
The University of Queensland
A closed network of queues:
All nodes have infinitely many servers:
, , where (<1) is proportional to the arrival rate at node j divided by service rate. Clearly for each n as , and so all nodes are bottlenecks.
All nodes have a single server:
The distribution of cannot be written down explicitly, but we can show that if there is a node j whose traffic intensity is strictly greater than the others, it is the unique bottleneck.
Moreover, for each node k in the remainder of the network, the distribution of approaches a geometric distribution with parameter in the limit as , and , for , are asymptotically independent.
Our only assumption:
The steady-state (joint) distribution of the numbers of customers at the various nodes has the product form
where S is the finite subset of with and is a normalizing constant chosen so that sums to unity over S.
Here is proportional to the amount of service requirement (in items per minute) coming into node j (this will actually be equal to ). Suppose (wlog) that .
is the service effort at node j (in items per minute) when there are n customers present. We shall assume that and whenever .
Our primary tool:
Define generating functions by
It is easily shown that , where takes the coefficient of a power series. The marginal distribution of can be evaluated as
Suppose that each node j has a single server ( for ). Then, and so . Summing
over n, and recalling that , gives .
Suppose that , so that node J has maximal traffic intensity.
If we can prove that as , then (node J is a bottleneck) and for j<J (the others are not).
WHY DOES ?
Define , where now . Clearly has radius of convergence (RC) ; in particular, ( ) has RC .
Claim: has RC for all i, so that
Proof: Suppose has RC and consider
Clearly , since , and so
implying that has RC .
THE GENERAL CASE
Message: Bottleneck behaviour depends on the relative sizes of the radii of convergence of the power series .
Proposition 1: Suppose has radius of convergence and that . Suppose also that
has a limit as . Then, node J is a bottleneck.
Example: Suppose node j has servers, so that the traffic intensity at node j is proportional to . Since , we have , and so . Therefore is proportional to the reciprocal of the traffic intensity at node j. It can be shown that (2) holds.
What happens when the generating functions corresponding to two or more nodes share the same minimal RC?
Proposition 2: In the setup of Proposition 1, suppose that and that for . Then, nodes behave jointly as a bottleneck in that as .
It might be conjectured that when the generating functions corresponding to two nodes share the same minimal RC, they are always bottlenecks individually. However, while this is true when all nodes have a single server (because ), it is not true in general.
Consider a network with J=2 nodes and suppose that . In the following examples and have the same RC .
Only one node is a bottleneck:< Suppose that and for . Then, it can be shown that and as .
Neither node is a bottleneck: Suppose that for . Then, as .
AND FINALLY ...
Proposition 4: Suppose that have the same strictly minimal RC , and that converges monotonically for some . Then, node 1 is a bottleneck if and only if
A sufficient condition for node 1 to be a bottleneck is that diverges at its RC and
This latter condition is not necessary: In the setup of the previous examples, suppose that and for . Then, and have common RC and both converge at their RC. But, it can be shown that is bounded above by a quantity which is as , implying that node 1 is a bottleneck.