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Mobility is an indispensable activity of our daily lives and road transport is one popular approach to mobility. However road congestion occurrence can be irritating and costly. This work contributes to the modeling and therefore predicting road congestion of a Ghanaian urban road by way of queuing theory using stochastic process and initial value problem framework. The approach is used to describe performance measure parameters, allowing the prediction of the level of queue built up at a signalized intersection as an insight into road vehicular congestion there and how such congestion occurrence can be efficiently managed.

Being the capital of Ghana, socio-economic factors in Accra are influencing rapid growth in vehicular population. Human population growth in the capital coupled with increased road vehicle ownership is the main factor fueling vehicular population increases. Human population growth in the capital is 4.4% above national average, and presently road vehicle ownership is increasing at 83.9% per annum [

Road traffic management (RTM) in Ghana, like most developing countries, has been very challenging and most attempts to address the problem have yielded little results [

However relatively little attention, by way of research, has been given to this congestion problem in the urban towns of Ghana. A scientific approach investigation of the problem could encourage the improvement of transport policies and strategies in place to mitigate this economic debilitating spate of congestion on our roads. For instance as a first step towards prescribing a solution to the problem, the ability to forecast traffic volumes for any time period, especially those critical periods, cannot be over emphasized. This forecast approach also has the capacity to directly support proactive traffic control, including educated deployment of TW to critical areas as well as accurate travel time estimations.

The primary contribution of this paper is to demonstrate a modeling of traffic evolution on an arterial road to Malam highway that serves surrounding suburbs and communities. Other possible benefit of this work is that it serves as a basis to other interesting investigations to characterize traffic congestion and the results obtained may serve as vital inputs to decisions that seek to improve traffic control and management. The objective therefore is to investigate the problem of congestion on the road segment and subsequently build upon this investigation to develop efficient tools capable of predicting and providing intelligent information on vehicular traffic flow.

Queues are waiting lines which occur whenever units must wait for a facility because the facility may be busy and therefore it is unavailable to render service required. The study of queues describes this phenomenon and since Erlang’s pioneering work for queuing theory, a number of authors have applied the theory in many areas (see [

This section gives the theory used in this work including brief general features of a queuing systems as well as the concept of stochastic process description of a Markovian queuing system from which time dependent transi- tions obtained using initial value problem approach and various performance measure characteristics deduced.

Typically any queuing system is composed of units, referred to as customers, needing some kind of service and who arrive at a service facility, join a queue if service is not immediately available, and eventually leave after receiving the service. A server refrerrs to mechanism that delivers service(s) to the customer. If upon arrival a “customer” finds the server busy, then s/he may form a queue, join it or leave the system without receiving any service even after waiting for some time [

1) Customers are vehicles using the road infrastruction at the signalized intersection for which the traffic light in use to regulate vehicular flow through the intersection is the server.

2) Vehicles arrive randomly as units and form a single file of waiting line until they are served by a server.

3) Vehicles are served individually in parallel, according to the order in which they arrived.

4) The Arival Pattern: This is the manner in which arrivals occur, indicted by the inter-arrival time between any two consecutive arrivals. For our stochastic modelling framework, the inter-arrival time may vary and may be described by a specific probability distribution that best describes the arrival pattern observed.

5) Arrival Rate

6) The Service Pattern: This is the manner in which the service is rendered and is specified by the time taken to complete a service. Similar to the arrival pattern, distribution of the service time must be specified under sto- chastic modelling considerations.

7) Service Time

8) Server Utilization

9) Mean Service Time

10) Mean Waiting Time T: The average time spent in the queue by a customer who receives a service.

11) Mean Queue Size N: The Average number of customers in the system for service

The quality of service one receives could be judged, at least in part, by the length of time one waits in the queue for service and this is very much influenced by what constitutes the configurations of the syetem. We can indicate this configuration as A/B/C/X/Y/Z, [

We let

1) Distribution of the number of vehicles

2) Distribution of the waiting time in the queue, the time that an arrival has to wait in the queue. Suppose

3) Distribution of the virtual waiting time

4) Distribution of the busy period being the length (or duration) of time during which the server remains busy. The busy period is the interval from the moment of arrival of a unit vehicle at an empty system to the moment that the channel becomes free for the first time. This therefore constitutes a random variable.

Suppose that we observe the state of the vehicular traffic at discrete time points t = 0, 1, 2,… for which suc- cessive time points define a set of random variables (RVs)_{n} assumes the finite set of values

Equation (1) indicates a relation of dependence between the RVs

Given that the state space

have a time homogeneous continuous Markov Chain. A discrete time Markov chain in which transitions can happen at any time is known as continuous time Markov chain [

Suppose that

where

miting probability that there are

If we consider two sets of limiting probabilities

For a queuing system of many vehicles and one server, suppose

time-dependent transition probabilities for this Markov chain, let

tion vector for the states “being served” and “finished being served” so that

distribution vector becomes

If we let

time step is given by

the corresponding matrix equation in (6).

Performing matrix multiplication then we have

Applying differential calculus and taking limits so that

and whose solution using Initial Value Prblem approach, is given by Equation (8b).

Since we assumed that there is an individual initially being served, then for initial conditions

Also since

Let T denote the time at which service is completed. This is considered to be the time taken for making a transi- tion from one state to another, then T is a continuous random variable with range

Thus the reciprocal of Equation (10f) gives

Suppose at each instant of time, the queue is in a certain state

If we let the traffic intensity

rate increases and approaches the mean service rate, the

Therefore substituting Equations (13b) into Equation (12), gives Equation (14) the probability that the queue is in state

Now to compute the mean queue size

Suppose the queue is empty (state 0) when an item arrives, then the server will begin processing the item im-

mediately, and the item will spend the mean service time

queue is in State 1 when an item arrives, the item will spend twice the mean service time in the queue before de-

parting and this will have a value of

In this section, we apply the theory to data we collected for the traffic flow through Dansoman junction, on Malam-Kaneshie urban road. The junction is signalised with traffic lights and so constitutes a queuing system. We consider the application to traffic stream of vehicular movement and in this case the road vehicles constitute our arrivals process, the traffic light signal at the intersection is the server and the controlled passage gives the service process, with the green light as the service mechanism. The population in this case is of the infinite type.

Day 1 | Day 2 | Day 3 | Day 4 | Day 5 | Day 6 | Sum | |
---|---|---|---|---|---|---|---|

07:30 - 07:35 | 106 | 140 | 170 | 16 | 59 | 87 | 578 |

07:35 - 07:40 | 102 | 90 | 85 | 84 | 140 | 99 | 600 |

07:40 - 07:45 | 82 | 73 | 90 | 112 | 88 | 120 | 565 |

07:45 - 07:50 | 88 | 66 | 64 | 180 | 69 | 113 | 580 |

07:50 - 07:55 | 60 | 44 | 120 | 69 | 75 | 36 | 404 |

07:55 - 08:00 | 72 | 55 | 60 | 68 | 51 | 50 | 356 |

Total | 510 | 468 | 589 | 529 | 482 | 505 | 3083 |

Vehicles/sec | 0.28 | 0.26 | 0.33 | 0.29 | 0.27 | 0.28 | 1.71 |

light duration when vehicles flow through the intersection, given by

We also found out from observation that each vehicle spends an average of 240 second to traverse a distance of 0.6 kilometres which accommodates 65 vehicles on the average. We therefore have the waiting time and the queue length as 240 seconds and 65 vehicles respectively.

Accordingly we obtained the average arrival rate for the period as

We then determine the service rate

served that the average service time

queue length and therefore waiting time that can spelling catastrophic condition such as those perpetual con- gestions observed during the morning rash hour period. Also note the observed queue size of 65 vehicles as well as an average waiting time of 240 seconds. These values respectively compares with the theoretical values of

For a fixed mean service rate, when

most of the time. Also the mean wait time

the queue is empty, the item will wait in the queue for one mean service time. As the load on the queue increases, the mean queue size and the mean wait time will increase. When items arrive in the queue as fast as there are departing ones then

We have in this work, measured the traffic flow at the Dansoman Junction of Malam-Kaneshie urban road during the morning rush hours and have demonstrated features of the queue built up at the signalised inter- section with data and modeled the traffic flow there as a M/M/1. We have shown that the current queue system will continue to develop heavy traffic, evident by the growing queue length and waiting time, during the peak hours. This obviously has quality of service (QoS) implications for the system at the moment evident from the traffic intensity estimates from the data collected. This work therefore gives insight into possible undesirable le- vels of vehicular traffic congestion and the obvious question is how to minimise or at least contain these unde- sirable levels in order to optimise waiting time at the intersection. Possible line of attack includes the use of

Time | Served Vehicles | Light Duration (sec) |
---|---|---|

:48 | 18 | 60 |

:51 | 19 | 62 |

:54 | 17 | 59 |

:57 | 18 | 63 |

:00 | 16 | 60 |

Total | 88 | 304 |

Average | 17.6 | 60.8 |

signal time adjustments or increase in the road infrastructure. We explore this in our further work from which we report on realistic approach of mitigating the problem thereby improving the QoS performance.