OPTIMAL ALLOCATION OF EFFORT 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-mntraffic)

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*

DefineR(m,n) to be the collection of (distinct) links used by type-mntraffic:

where is the number of links on route and 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

iis 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 linkjis 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 ,