Bottlenecks in Markovian Queueing Networks
by
Phil Pollett
Department of Mathematics
The University of Queensland
OUR SETTING
A closed network of queues:
BOTTLENECKS
Common sense:
SIMPLE EXAMPLES
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.
MARKOVIAN NETWORKS
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
.
GENERATING FUNCTIONS
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
for .
SINGLE-SERVER NODES
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.
COMPOUND BOTTLENECKS
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.
SOME EXAMPLES
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.