RESOURCE ALLOCATION IN GENERAL QUEUEING NETWORKS
by
Phil Pollett
Department of Mathematics
The University of Queensland
PACKET SWITCHING NETWORKS
A packet switching network with 4 nodes (labelled A,B,C and D) and 5 links (labelled 1,2,...,5)
PACKET SWITCHING NETWORKS
N switching nodes (labelled 
)
J links (labelled 
)
Poisson traffic on route 
at rate 
(type-mn traffic)
Common expected message length: 
(bits)
(Message lengths have an arbitrary distribution which does not depend on type)
Transmission rate on link j is 
(bits/sec.)
(There is a first-come first-served (FCFS) discipline at each link)
ROUTING MECHANISMS
Fixed routing
Define R(m,n) to be the collection of (distinct) links used by type-mn traffic:
where
is the number of links on route
is the link used at stage s.
Random alternative routing
This can be accommodated within the framework of fixed routing by allowing a finer classification of type (mni):
where
is the probability that alternative route i is chosen; there is fixed set of alternative routes for each OD pair (m,n).
PACKET SWITCHING NETWORK AND CORRESPONDING QUEUEING NETWORK
A NETWORK OF QUEUES
Links 
queues
Messages customers
T - set of customer types
- arrival rate of type-t customers
(Independent Poisson streams)
Route for type-t customers:
A NETWORK OF QUEUES
If message lengths have an exponential distribution the links behave independently (indeed, as if they were isolated), each with independent streams of Poisson offered traffic (independent among types). For example, if
so that the arrival rate at link j is given by
and the demand by 
(bits/sec), then,
if the system is stable ( 
for each j), the expected
number of messages at link j is
and the expected delay is
THE INDEPENDENCE ASSUMPTION
Kleinrock (1964) proposed the following assumption: that successive messages requesting transmission along any given link have lengths which are independent and identically distributed, and that message lengths at different links are independent.
Thus, we shall assume that at link j message lengths have a
distribution function 
which has mean 
and
variance 
.
Even under this assumption, the model (now a network of 
queues with a FCFS discipline) is not analytically tractable.
To make progress, we shall use the Residual-life Approximation
(Pollett (1984)).
THE RESIDUAL LIFE APPROXIMATION
Let 
be the distribution function of the queueing
time at link j: the time a message spends in the buffer
before transmission. The Residual-Life Approximation
(RLA) provides an accurate approximation for :
where
and 
denotes the n-fold convolution of 
. The
distribution of 
, the number of messages at link j,
used in (2)
is that of the corresponding quasireversible network
of symmetric queues obtained by imposing a symmetry
condition at each link j. In the present context, this
amounts to replacing FCFS by
a last-come first-served (LCFS) discipline.
THE RESIDUAL LIFE APPROXIMATION
One immediate consequence of (2) is that the expected queueing
time 
is approximately
where 
is the expected number of messages at link j in the
quasireversible network. Hence, the expected delay at link j is
approximated as follows:
In the RLA, it is only which changes
when the service discipline is altered. For the present FCFS discipline
is given by
OPTIMAL ALLOCATION OF EFFORT
We shall minimize the average network delay, or equivalently the average number of messages in the network:
(using the RLA for 
).
$F - overall network budget
($-seconds/bit)
- cost of operating link j
(The cost of operating link j is proportional to the capacity
Thus, we should choose the capacities subject to the cost constraint
THE PROBLEM
Let 
be the squared coefficient of variation of
and let 
.
Introduce a lagrange multiplier 
; our
problem then becomes one of minimizing
Setting 
yields a quartic
polynomial equation in :
where 
.
Find solutions
such that 
(recall that this latter condition is a
requirement for stability).
Using the transformation
the problem reduces to one with unit costs 
;
equation (6) becomes
and the constraint becomes
EXPONENTIAL SERVICE TIMES
If transmission times are exponentially distributed ( 
for
each j) it is easy to verify that (10) has a unique solution on
given by
Upon application of the constraint (12) we arrive at the optimal capacity assignment
for unit costs. In the case of general costs this becomes
after applying the transformation (8). This is a result obtained by Kleinrock (1964).
THE GENERAL CASE
We shall adopt a perturbation approach, assuming that the lagrange multiplier and the optimal allocation take the following forms:
where by 
we mean terms of order 
.
The zero 
order terms come from Kleinrock's solution:
where
FIRST-ORDER SOLUTION
On substituting (14) into (10) we obtain an expression for
in terms of 
, which in turn is calculated
using the constraint (12) and by setting 
(the Kronecker delta).
To first order, the optimal allocation is
where
SENSITIVITY
Let , , be the new optimal allocation obtained
after incrementing 
by a small quantity 
. We find
that, to first order in 
,
and, for 
,
