Bottlenecks in Markovian Queueing Networks
by
Phil Pollett
Department of Mathematics
The University of Queensland
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:
is the number of customers at node j, then this node is a
bottleneck if, for all 
, 
as
.
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.
