Title:

Discrete time models for the time dependent behaviour of queues in series.

This work is funded by the British Council and Tanzania government with the overall
aim of undertaking research on queueing models that can be of practical use in the
management of queues in Tanzanian banks. The sponsorship required that the field
work be carried out in Tanzania.
The thesis has four mam parts namely: problem identification" exact model
formulation, development and evaluation of approximation models and demonstration
of use ofapproximation models.
Identification of the problem is through the field work. Field work took place over a
six month period in one of the Tanzania's commercial banks, the National Bank of
Commerce (NBC). Out of this field work the problem to study and the nature of the remaining research is identified. It falls under tandem queueing model with time
dependent arrival process.
The methodology chosen was based on that of Brahimi (1990) who has studied in
detail the approximation of single station queueing systems in discrete time with time
dependent arrival process. This work is an extension to his work but in tandem
queueing systems that are relevant in Tanzanian banks.
The modelling started with a simple tandem system of two nodes each with one server
and arrivals dependent upon time. An exact discrete time model was developed for
this situation. It is a new model but is limited to handling small capacity problems.
This prompted the need for approximation models.
Three approximation models are then developed. The difference between the
approximation models lies in the way in which they approximate the transfer of
customers between the nodes.
(i) Approximation I is a multiserver multinode model. At each epoch it models the
transfer probability as an aggregate of departure probabilities.
(ii) Approximation II is a tandem system of two single server nodes. The transfer
process is approximated using the distribution of the interdeparture (known in this
work as intertransfer) times. The approximation takes into account the actual state of
node one whenever a transfer is due. Whenever a transfer from node one happens the
approximation samples the distribution of the time to the next transfer. It will sample remaining research is identified. It falls under tandem queueing model with time
dependent arrival process.
The methodology chosen was based on that of Brahimi (1990) who has studied in
detail the approximation of single station queueing systems in discrete time with time
dependent arrival process. This work is an extension to his work but in tandem
queueing systems that are relevant in Tanzanian banks.
The modelling started with a simple tandem system of two nodes each with one server
and arrivals dependent upon time. An exact discrete time model was developed for
this situation. It is a new model but is limited to handling small capacity problems.
This prompted the need for approximation models.
Three approximation models are then developed. The difference between the
approximation models lies in the way in which they approximate the transfer of
customers between the nodes.
(i) Approximation I is a multiserver multinode model. At each epoch it models the
transfer probability as an aggregate of departure probabilities.
(ii) Approximation II is a tandem system of two single server nodes. The transfer
process is approximated using the distribution of the interdeparture (known in this
work as intertransfer) times. The approximation takes into account the actual state of
node one whenever a transfer is due. Whenever a transfer from node one happens the
approximation samples the distribution of the time to the next transfer. It will sample from the combined distribution of residual interarrival and service times at node one
if the number of units the latest transfer leaves at node one is zero. Ifthe latest transfer
leaves node one with one or more units, then, the distribution of time to the next
transfer is sampled from the service time distribution.
(iii) Approximation ill goes farther than approximation II in that it adds an extra
variable. This extra variable involves the correlation of the intertransfer times. In
approximation II the number of units left at node one by the latest transfer is used to
select the next intertransfer time. In this approximation the system sizes at node one
when the latest and the previous transfers happened are used to select the next intertransfer
time. Thus some element of the correlation between consecutive intertransfer
times is incorporated.
Approximation I is shown to have a good accuracy when the service time distribution
at node one is close to negative exponential in shape. Approximation II has good
accuracy for high traffic intensities irrespective of the distribution of service time at
node one. However for medium traffic intensities at node one and near exponential
service time at node one, it is not as good as approximation 1. In general
approximation III is shown to be better than approximations I and II, although there
are some exceptions when, for near negative exponential service time at node one,
approximation I is still a little better.
The potential of using the discrete time tandem queueing models with time dependent
arrival process for problems such as those faced in Tanzanian banks is demonstrated using approximation I with some of the NBC data Several scenarios that highlight
different performances are produced. These scenarios investigate different managing
strategies for the queueing systems with this type ofqueueing problem.
The thesis ends by pointing out areas for further research in order to improve and
further develop the scope of the approximations considered in this work.
