Mathematical Theory and Modeling www.iiste.org ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online) Vol.4, No.9, 2014 31 Central Moments of Traffic Delay at a Signalized Intersection Benard Ronoh 1* , Kennedy Nyogesa 2 , Argwings Otieno 1 , Betty Korir 1 1. Department of Mathematics and Computer Science, University of Eldoret, P.O Box 1125, Eldoret, (Kenya) 2. Department of Mathematics, Masinde Muliro University of Science & Technology, P.O Box 190, Kakamega (Kenya) * E-mail of the corresponding author: ardyben@gmail.com The research is financed by National Council of Science & Technology (4 th Call MSc. Grant 2011/2012) Abstract Traffic delay model was developed under a basis of deterministic and stochastic delay components. The latter component was put under D/D/1 framework and therein mean and its variance derived. While the stochastic component was put under the M/G/1 framework, mean and variance derived. Extension on stochastic component and M/G/1 framework was discussed with the usage of compressed queueing processes. Harmonization of the moments of deterministic and stochastic components to obtain the overall central moments of traffic delay has been discussed. Simulation was performed using Matlab for traffic intensities ranging from 0.1 to 1.9. The simulated results indicate that both deterministic and stochastic components are incompatible as the traffic intensity approaches capacity. Furthermore the results indicated that oversaturated conditions and random delay renders the stochastic component in traffic delay models unrealistic. Also, with the ability to estimate the variance of overall traffic delay, it is feasible to integrate the concept of reliability into design and analysis of a signalized intersection. Keywords: D/D/1 model, M/G/1 model, compressed queueing processes, simulation. 1.0 Introduction Traffic delays and queues are principal measures of performance that determine the level of service (LoS) at signalized intersections. They also evaluate the adequacy of the lane lengths and the estimation of fuel consumption and emissions. Quantifying these delays accurately at an intersection is critical for planning, design and analysis of traffic lights. Signalized intersection referred herein, is a road junction controlled by a traffic light. Traffic lights were implemented for the purpose of reducing or eliminating congestions at intersections. These congestions exist because an intersection is an area shared among multiple traffic streams, and the role of the traffic light is to manage the shared usage of the area. Traffic models in an intersection are always subjected to both uniform and random properties of traffic flows. As a result of these properties, vehicle travel times in an urban traffic environment are highly time dependant. Models that incorporate both deterministic and stochastic components of traffic performance are very appealing in the signalized intersection since they are applied in a wide range of traffic intensities as well as to various types of traffic lights. They simplify theoretical models with delay terms that are numerically inconsequential. Of the various queueing models, D/D/1 and M +Δ /G +Δ /1 were used in this study. The D/D/1 model assumed that the arrivals and departures were uniform and one service channel (traffic light) existed. This model is quite intuitive and easily solvable. Using this form of queueing with an arrival rate, denoted by  and a service rate, denoted by  , certain useful values regarding the consequences of queues were computed. The M +Δ /G +Δ /1 model used implied that the vehicles arrived at an intersection in a Poisson process with rate  and were treated in the order of arrival with inter arrival times following exponential distribution with parameter  . The service times were treated as independent identically distributed with an arbitrary distribution. Similarly, one service channel (traffic light) was considered in this model. 2.0 Traffic Problem The main traffic problem here is to develop overall traffic delay model using D/D/1 and compressed M +Δ /G +Δ /1 queueing systems. With D/D/1 system, the inter-arrivals and service times are deterministic while in M +Δ /G +Δ /1 system, the Markovian arrivals and iid service times follow a general distribution. A single service channel