Bottlenecks in Markovian Queueing Networks


Phil Pollett

Department of Mathematics
The University of Queensland


A closed network of queues:

Examples: Can we identify regions of congestion (bottlenecks) from the parameters of the model?


Common sense:

The nodes with the smallest service effort will be the most congested.

A formal definition:
If tex2html_wrap_inline649 is the number of customers at node j, then this node is a bottleneck if, for all tex2html_wrap_inline653 , tex2html_wrap_inline655 as tex2html_wrap_inline657 .


All nodes have infinitely many servers:

tex2html_wrap_inline659 , tex2html_wrap_inline661 , where tex2html_wrap_inline663 (<1) is proportional to the arrival rate at node j divided by service rate. Clearly tex2html_wrap_inline669 for each n as tex2html_wrap_inline657 , and so all nodes are bottlenecks.

All nodes have a single server:

The distribution of  tex2html_wrap_inline649 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 tex2html_wrap_inline681 approaches a geometric distribution with parameter tex2html_wrap_inline683 in the limit as tex2html_wrap_inline657 , and tex2html_wrap_inline681 , for tex2html_wrap_inline689 , are asymptotically independent.


Our only assumption:

The steady-state (joint) distribution tex2html_wrap_inline691 of the numbers of customers tex2html_wrap_inline693 at the various nodes has the product form


where S is the finite subset of tex2html_wrap_inline699 with tex2html_wrap_inline701 and tex2html_wrap_inline703 is a normalizing constant chosen so that tex2html_wrap_inline691 sums to unity over S.

Here tex2html_wrap_inline663 is proportional to the amount of service requirement (in items per minute) coming into node j (this will actually be equal to tex2html_wrap_inline713 ). Suppose (wlog) that tex2html_wrap_inline715 .

tex2html_wrap_inline717 is the service effort at node j (in items per minute) when there are n customers present. We shall assume that tex2html_wrap_inline723 and tex2html_wrap_inline725 whenever tex2html_wrap_inline727 .


Our primary tool:

Define generating functions tex2html_wrap_inline729 by


It is easily shown that tex2html_wrap_inline733 , where tex2html_wrap_inline735 takes the tex2html_wrap_inline737 coefficient of a power series. The marginal distribution of tex2html_wrap_inline649 can be evaluated as


for tex2html_wrap_inline743 .


Suppose that each node j has a single server ( tex2html_wrap_inline747 for tex2html_wrap_inline727 ). Then, tex2html_wrap_inline751 and so tex2html_wrap_inline753 . Summing


over n, and recalling that tex2html_wrap_inline733 , gives tex2html_wrap_inline761 .

Suppose that tex2html_wrap_inline763 , so that node J has maximal traffic intensity.

If we can prove that tex2html_wrap_inline767 as tex2html_wrap_inline657 , then tex2html_wrap_inline771 (node J is a bottleneck) and tex2html_wrap_inline775 for j<J (the others are not).

WHY DOES tex2html_wrap_inline779 ?

Define tex2html_wrap_inline781 , where now tex2html_wrap_inline783 . Clearly tex2html_wrap_inline785 has radius of convergence (RC) tex2html_wrap_inline787 ; in particular, tex2html_wrap_inline789 ( tex2html_wrap_inline791 ) has RC tex2html_wrap_inline793 .

Claim: tex2html_wrap_inline795 has RC tex2html_wrap_inline797 for all i, so that


Proof: Suppose tex2html_wrap_inline803 has RC tex2html_wrap_inline805 and consider


Clearly tex2html_wrap_inline807 , since tex2html_wrap_inline809 , and so


implying that tex2html_wrap_inline813 has RC tex2html_wrap_inline815 .


Message: Bottleneck behaviour depends on the relative sizes of the radii of convergence of the power series tex2html_wrap_inline729 .

Proposition 1: Suppose tex2html_wrap_inline785 has radius of convergence tex2html_wrap_inline821 and that tex2html_wrap_inline823 . Suppose also that


has a limit as tex2html_wrap_inline825 . Then, node J is a bottleneck.

Example: Suppose node j has tex2html_wrap_inline831 servers, so that the traffic intensity at node j is proportional to tex2html_wrap_inline835 . Since tex2html_wrap_inline837 , we have tex2html_wrap_inline839 , and so tex2html_wrap_inline841 . Therefore tex2html_wrap_inline821 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 tex2html_wrap_inline847 and that tex2html_wrap_inline849 for tex2html_wrap_inline851 . Then, nodes tex2html_wrap_inline853 behave jointly as a bottleneck in that tex2html_wrap_inline855 as tex2html_wrap_inline657 .

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 tex2html_wrap_inline859 ), it is not true in general.


Consider a network with J=2 nodes and suppose that tex2html_wrap_inline863 . In the following examples tex2html_wrap_inline865 and tex2html_wrap_inline867 have the same RC tex2html_wrap_inline869 .

Only one node is a bottleneck:< Suppose that tex2html_wrap_inline871 and tex2html_wrap_inline873 for tex2html_wrap_inline727 . Then, it can be shown that tex2html_wrap_inline877 and tex2html_wrap_inline879 as tex2html_wrap_inline657 .

Neither node is a bottleneck: Suppose that tex2html_wrap_inline883 for tex2html_wrap_inline727 . Then, tex2html_wrap_inline887 as tex2html_wrap_inline657 .


Proposition 4: Suppose that tex2html_wrap_inline891 have the same strictly minimal RC tex2html_wrap_inline893 , and that tex2html_wrap_inline717 converges monotonically for some tex2html_wrap_inline897 . Then, node 1 is a bottleneck if and only if


A sufficient condition for node 1 to be a bottleneck is that tex2html_wrap_inline865 diverges at its RC and


This latter condition is not necessary: In the setup of the previous examples, suppose that tex2html_wrap_inline871 and tex2html_wrap_inline911 for tex2html_wrap_inline727 . Then, tex2html_wrap_inline865 and tex2html_wrap_inline867 have common RC tex2html_wrap_inline869 and both converge at their RC. But, it can be shown that tex2html_wrap_inline921 is bounded above by a quantity which is tex2html_wrap_inline923 as tex2html_wrap_inline657 , implying that node 1 is a bottleneck.