sDISTRIBUTION OF GAPS AND BLOCKS CD IN A TRAFFIC STREAM

This paper studies some of the theoretical questions of large openings or gaps in a single stream of traffic. A gap in the traffic stream is defined as a headway between vehicles greater than or equal to some minimum size -say x. Several authors have studied the probability distribution of the wait which a randomly located observer must endure before he finds a gap. This paper, while briefly reviewing the solutions of this well known problem, is primarily concerned with expressions for: (i) the distribution of gap sizes; (ii) the distribution of spacings between vehicles and gaps; (iii) the mean and variance of inter-vehicle and inter gap spacings; (iv) the stationary flow rates of gaps; and (v) the distribution of blocked and unblocked periods. It is assumed that the origin of measurements may be located (i) with the passing of a vehicle, (ii) at the beginning of a gap. or (iii) at random. It is also assumed that the distribution of inter-vehicle spacings are independently, but identically, distributed random variables. • k 0. Introduction liiere are several well-known variations of a merging problem which arise when traffic in a minor stream joins or crosses a major flow stream, the criterion for merging being that a minimum size gap appear between vehicles in the major stream. As the problem is usually formulated one or all vehicles in the minor stream merge with the major stream whenever the headway to the next vehicle is greater than or equal to the minimum size gap. * In automobile traffic this problem arises in low-speed merges when cars come to an intersection, stop and wait for the desired gaps in a cross stream of traffic. While this situation may be one of the more obvious places where a study of gaps in traffic streams is needed there are many other examples: In high-speed automobile merges the question is often one of finding empty road-space into which cars can fit. At airports departing aircraft wait for the use of a runway which also services the higher priority landing aircraft; the former can only occupy the runway when gaps of certain minimum size appear between successive landing aircraft. Flight rules often specify for safety and/or reasons of dotectability that, in crossing air-lanes, aircraft at the same altitude must at all times be separated by some minimum headway from one another. While I have introduced the gaps in the major stream in the context of the wait of a vehicle in the minor stream which looks for the first gap or opening in the major stream, it is at this point that 1 would like to make clear the distinction between the major and minor stream and clearly identify the fact that the gap production process is one which can go on in the major stream in the absence of a minor stream of crossing or merging traffic. *In earlier papers the word "gap" has often referred to any empty interval; in this paper "gap" specifically refers to intervals between vehicles greater than or equal to x, the minimum size gap. The word "minor" certainly seems appropriate when one considers mathematical merging models which allow at most one merging vehicle per gap into the major stream^ In this case it is easy to show that the stationary flow rate of gaps is always less than the stationary flow rate of vehicles in the major stream; hence the phrase jpinor stream refers to one whose steady state flow rate is less than that of the major stream. However, in actual merging situations it is often the case that the "minorstream contributes the larger fraction of downstream flow. Generally speaking, ae vehicle flow rates in the major stream decrease, gap sizes increase; many vehicles in the minor stream may be absorbed by a single large gap in the major stream. If. on the average, the total rate of absorption is greater than the -major" stream flow rate it would seem natural to reverse the names of minor and major stream. It is this apparent contradiction which has led me to consider the role of these two streams and to think perferably in terms of a primary and a secondary flow process. This paper will be primarily concerned with the statistical characteristics of large gaps or openings in a single traffic stream. The flow of vehicles in this major stream, though unpredictable, does not depend on the presence or condition of vehicles in an intersecting or merging stream. Large openings appear in the major stream only because the statistics cf inter-vehicle spacings in that stream tell us that one can expect to find vehicles separated by gaps a certain fraction of the time. The casual reader of earlier papers might come to the conclusion that the gap-producing process is a function of the wait endured by a vehicle in the minor stream; rather, it is the other way around, the gap process in the major stream and a specific merging, stopping or crossing mechanism helps to generate a secondary or minor stream process. The approach used in this paper differs from earlier ones in that the statistical characteristics of the gap-producing process in the major stream are examined and then used to analyze a secondary process (of which there may be many). It is important to point out, nonethaless, that the observation of ihe queue length of vehicles or the wait of a single vehicle in a minor stream may be of immense predictive value in determining the appearance of gaps in the major stream. Assume that inter-vehicle spacings are independently and identically distributed. A certain fraction of the vehicles will be separated by intervals greater than or equal to x while the remainder will be separated by intervals less than x. In Section (1) expressions are obtained for the probability distribution of counts of gaps when the distribution of headways between vehicles or gaps is not included in the discussion. In Section (2) the probability law which describes the length of the interval from the beginning of a gap to the beginning of a successive gap is found as a function of the inter-vehicle distribution and the minimum gap size. The mean and variance of this inter-gap distribution are then related to the mean and variance of the distribution of inter-vehicle headways. In Section (3) expressions are obtained for the distribution of the size of blocked* and unblocked periods and the wait for the first unblocked period. In Section (4) the discrete gap counting distributions and the average and variance of the gap and block flow rate are derived. In Section (5) analytic and numerical results are given for the case where intervehicle headways are exponentially distributed, i.e., vehicle counts are Poisson. In Section (6) moments of the inter-evsnt distributions are obtained in terms of the vehicle flow rate when vehicles appear in bunches but each vehicle is separated by a minimum headway. Section (7) compares some of the results obtained in this paper with results of earlier authors. A blocked period is one in which the headway for any point to the next vehicle to appear is less than or equal to x, the minimum gap. See Figure (1) and paragraph preceding Equation (3.1), in this paper and Figure (1) in Reference (11).