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PDF (DX094490.pdf) - White Rose Etheses Online

PDF (DX094490.pdf) - White Rose Etheses Online

25 exceeded by a

25 exceeded by a considerable margin. In fact zero delays are predicted until demand reaches capacity, contrary to experience. Therefore, both sets of models perform worse at the region when capacity and demand are equal or of similar value which, Kirnber and Hollis observe, in practical terms is the most important region of operation. They proceeded to develop an alternative model based on time-dependent demand-capacity interaction. They define p as the capacity. and q the demand flow. They assume that these values vary in time, and that they represent average values at each fraction of the period of interest. Each section of this period re presents a possible set of arrivals to the queue and departures from it. The proportion of occurrences of a queue of n vehicles at time t is p(t). Both the average queue length and average vehicular delay can be derived as functions of time from p(t). Hollis, Semmens and Denniss (1980) report on a computer program to model capacities, queues and delays at roundabouts which is based on an approximate method of the above principle. This employs a co-ordinate transformation technique to smooth the steady state stochastic relationship for queue length or vehicle delay into the over-capacity deterministic results obtained by integrating the excess of demand over capacity. An example of a graph is given in Fig. 2.10. The queue lengths and delays are calculated according to the following rules: Over a short time interval, t, with capacity p and demand q assumed constant, traffic intensity is defined as p = q/p. Several cases exist depending on P' the queue at the start of the time interval L 0 , and the equilibrium queue length

£ = p /( l - p). variable) by: 26 If is a queueing function defined for x (a time = 0.5 {((px(1-p) +1)2 +. 4ppx) ½ - (px(1-p) + i)} (eq.2.24 then the average queue length, L, after a time, t,is given by the following expressions: (j) for p ^ 1: L(t) = (ti-t ) where t = L (L+1)/ n 0 0 0 0 i(p(L0+1) - L0) (ii) for p < 1: (a) 0 ^ L0 < £:L(t) = n(t+toJ (b) L0 = 9,: L(t) where t =L (L+1)/i(p(L+1)-L 0 0 0 0 0 (c) 9.. < L ^ 29.: L(t) = 22, - E(t+t0) 0 where t0=(2Z-L0 ) (22,-L 0+ 1) /p (p(22,-L0+ 1)- (29.,-L) (d) L > 22. L + (p - L /(L + 1))j.it 0 ^ t ^ t 0 0 0 0 C :L(t) 29. - F (t-t ) t > t n C C where t = (29.-L )/p(p - L /(L +1) C 0 0 0 These equations represent the growth or decay in queue length within the time interval t. The total average delay during this time is obtained by integrating the approp- riate queue length equation over the time interval. Thus, given the demand flow, q, and capacity, i for a short time interval and the queue length at the beginning of the interval, the equations above allow the queue length at the end of the interval to be calculated. Therefore if any period is divided into a sequence of short time intervals the

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