APPENDIX - A

A BRIEF DESCRIPTION OF A PROGRAM TO MODEL AN IRT

We have instituted a computer program to study control and capacity on an Individual Rapid Transit system. The model we have adopted makes no attempt to consider the various physical actions required, for instance, to actual launch a carrier on to the system. Nor does it account for the time actually in-station between the time a particular vehicle presents itself to the system until it is launched. For that matter, the time actually involved in travel between stations is also of no concern.

The model is a statistical one. Thus, the probability of a particular set of circumstances occurring at a particular location, at any given time is identical to the same set of circumstances happening at any other time. Thus the fact that the packet takes a finite time to go from one station to another is of no consequence; the probability that it will encounter a particular set of circumstances is unchanged. It is the sequence of events that is important.

The general approach is to systematically vary the number of vehicles entering the system, together with each vehicle’s destination. for each entry station. A packet simulates a journey throughout the entire system and reacts appropriately with the several stations to whatever are this cycle’s numbers. For each cycle, a record is maintained of the critical parameters of the system. These include the number of vehicles in the packet as it moves along the system, the number getting on or off at each station and the like. The process is repeated in the next cycle with a new set of numbers.

This is repeated for upwards of a 100 thousand cycles, each time with a new set of numbers. This continues until the normalized distribution of system parameters does not change appreciably. For instance the fraction of times the packet contains16 vehicles is essentially constant, or the average hourly rate of vehicles departing at a specific station does not change. This general approach is known as a Monte Carlo technique.

By this method, we obtain a statistical picture of the operation of the system. We can not tell exactly when a particular event will happen; but we can determine with some certainty how often it is likely to occur.

In the instant case we have chosen Interstate 405, the San Diego Freeway as the basis for traffic demand. Caltrans (California Department of Transportation) freeway data comes in two flavors. One is “1998 traffic volumes on california state highways” which provide data on traffic volume between specified points. Unfortunately, for our purposes, the data combines traffic in both directions.

The other is “1998 Ramp Volumes On the California State Freeway System”. This latter provides specific ramp data, but only in terms of average daily traffic (ADT).

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Thus to have a consistent set, we were forced to do so in terms of the average daily traffic. We made the assumption that averaged over 24 hours; the traffic in one direction was equal to the other. That is, we divided the average freeway traffic by two. The algebraic sum of the ramp data did not precisely equal the halved freeway data. Thus we felt compelled to adjust the data to force consistency. For the most part, we adjusted the ramp data between specified points to conform to freeway data. We did this under the, perhaps doubtful, assumption that freeway traffic is counted more often, and thus should be more accurate. Moreover, an error in ramp data perpetuates along the entire downstream system. At any rate, we did this by maintaining the relative contribution of intervening ramps constant; adjusting the absolute value in proportion to the initial values. In this way we obtained a self-consistent set.

To obtain destination probabilities, we assumed that the traffic was completely homogenized. That is, if at some point along the system the average departure rate to a specific off-ramp is known; and the average total traffic is known; then the fraction departing at this station can be calculated.

Further, assume that the contribution to the total traffic from station A is, say, 10 percent. Thus, if the average departure rate is 30 percent of the total, we assume that the average contribution from station A is 3 percent. Remember that, we already know what the average traffic at this location, as well as the number leaving. It remains to be calculated what station A’s contribution is.

We start at the first eligible departure station for vehicles entering at station A. At this point we know what its contribution to the total traffic is; as know the average input for station A, and at this point none have departed. Thus we can calculate station A’s contribution.

We subtract this contribution from the initial value, and proceed to the next departure station, where we repeat the process. We continue in this manner to the end of line. The probability of that any vehicle entering station A will wish to depart at any station along the line is simply the average number departing at that station divided by the average number entering. In both instances, referring to station A.

Note that we indicated the first eligible departure station. We assume that, for the most part, people will not get on the freeway only to depart at the next off-ramp. In an IRT we can enforce this. Thus, for normal on-ramps the probability for departure at the next ramp is set to zero, unless it happens to be a freeway off-ramp. In this case, the normal calculations apply. The same is true for a vehicle entering from a freeway on-ramp; the next ramp is always a valid off-ramp.

In this manner we build a complete probability profile for each entry station. We do not claim, necessarily, a high degree of accuracy with actual origin-destination relationships. We do argue that the results are consistent with actual data, and as a minimum, this approach can be defended as logically consistent. Moreover, our objectives are to study the total capacity of the system, along with the probability of conflicts, and the distribution of wait-cycles to gain entry. These are principally due to the statistical variations of on- and off-ramps. In this, we are in accord.

Having accomplished this, we still lack knowledge for how many vehicles will enter the system each cycle, along with the destination. For this, we require a method to assign specific values. It is most certainly not a uniform distribution from minimum to maximum.

Lets, first, consider the number entering during a given cycle. We assume that over the short run, during each cycle, cycle to cycle, the entries are random. Under this assumption, we can use Poisson statistics. In fact we must use these, as we are restricted to small number and integers (i.e., no fractional vehicles). The probability that a given number, n, will enter during a single cycle is then given by:[1]

p _n = (r ⁿ / n!) exp (-r)

where r is the mean or average value. Note that r can take on any value, but n is restricted to integers.

If we apply random numbers to what amounts to a table of values reflecting this probability distribution, we can effect the appropriate distribution. Over the short term, the results look completely random; considered over a large number of samples; the distribution of vehicles entering each station mirrors this distribution.

An example of the output from this process is given below. The specified mean value is 6.3, and the number of iterations, or samples, is one hundred thousand. As one can see the comparison is not identical with the theoretical value, but that is the nature of any random process. The numbers in the real world won’t be exact either; we are dealing in probabilities which, by definition, for a finite sample is not exact.

Table AI – Random Entry with a Poisson Distribution

and a Specified Mean of 6.3

N	Entry with N Vehicles	Normalized Number	Theoretical Value
0	187	1.8700e-003	1.8363e-003
1	1143	1.1430e-002	1.1569e-002
2	3715	3.7150e-002	3.6441e-002
3	7473	7.4730e-002	7.6527e-002
4	11999	1.1999e-001	1.2053e-001
5	15491	1.5491e-001	1.5187e-001
6	15538	1.5538e-001	1.5946e-001
7	14438	1.4438e-001	1.4352e-001
8	11295	1.1295e-001	1.1302e-001
9	8006	8.0060e-002	7.9113e-002
10	5107	5.1070e-002	4.9841e-002
11	2840	2.8400e-002	2.8545e-002
12	1469	1.4690e-002	1.4986e-002
13	740	7.4000e-003	7.2626e-003
14	319	3.1900e-003	3.2682e-003
15	140	1.4000e-003	1.3726e-003
16	58	5.8000e-004	5.4047e-004
17	31	3.1000e-004	2.0029e-004
18	8	8.0000e-005	7.0103e-005
19	3	3.0000e-005	2.3245e-005
20	0	0.0000e+000	7.3220e-006

Average: 6.3093

In a similar manner, using the previously described destination data, we can generate a distribution of individual vehicles departing to specific destination that reflects these probabilities.

Thus, for each entry station and for each cycle, two separate random calculations are made. One to determine the specific number entering the station; and two, for each vehicle entering a specific destination is assigned. Each vehicle is entered into a station enter-queue to await subsequent launching onto the system.

As we indicated, a large number of cycles, upwards of a 100 thousand or more, are simulated. For each cycle, at each entry station representing a freeway connecting on-ramp, vehicles are added without regard to quota restrictions. The only exception is when the addition of an additional vehicle would exceed the maximum allowed any packet. If this condition occurs, i.e., a vehicle is denied entry, it is recorded as a conflict.

At ordinary on-ramp entry station, a system of quotas restricts entry. There are two types of quotas. One is a restriction on the number in a given packet that can exit at a specific station. This is primarily used at freeway exits to insure that a particular cycle does not overload the capacity of the receiving freeway to accept.

The other is what is called a continuing-on quota; that is a restriction on the number of vehicles that can continue on beyond a specific location. Thus a vehicle might have to satisfy several of these quotas on an extended journey. This is primarily used to insure that vacancies do occur at incoming freeway on-ramps. Failure of any will result in deferred entry. Satisfying both quota results in deleting the vehicle from the enter-queue and entering it into what amounts to a packet-queue. That is, the vehicle is launched.

Obviously, in addition to entry station, there are also exit stations allowing vehicles depart. This is registered in the program by deleting the vehicle from the packet queue.

At freeway exits, for each cycle, we make a statistical estimate of the number of vehicles in the receiving packet. As with freeway entrances, if the receiving packet were to be oversubscribed, this also is recorded as a conflict, and entry of the guilty vehicle to the new line is deferred.

These estimates of the receiving packet are made starting with the published data on the average traffic at the specific entry point. As with the line under consideration (in this instance the 405), a continuing-on quota is established for the new line. This is based on the average traffic flow at that location. A Poisson calculation is made. However if the number exceed the quota, only the quota amount is allowed to continue. Similar to an entry station the remainder is placed in a queue for entry to a subsequent packet.

Recognizing that circumstances can dictate that the quota is occasionally exceeded, the possibility of up to 3 additional vehicles in the packet is provided for. Each of these is controlled by an input value. That is for one additional vehicle, a probability is input. For two, a smaller probability is specified; and for three, a similar input. Application is random consistent with the specified probabilities. These additional vehicles may, or may not result in exceeding the quota. That is, the continuing-on quota for this location on this line. In either instance, they are added to the calculated traffic unless that would exceed the maximum for any packet.

It should be noted that in a comprehensive calculation applying to an entire system, only the initial trial would require this provision. In subsequent calculations the calculated probabilities derived from that specific line (i.e., the receiving line) would be used.

Similar allowance for added vehicles is provided for freeway entries. As with departures, this provision would only be required for the first trial. .

Thus we can study the reaction of the system as we change the number applying to the system, and the various quota restriction we apply. As we indicated previously, changes to the number applying are effected universally. That is, a single number multiplies the average daily traffic at each location.

As indicated, a cumulative record of the various parameters affecting control and capacity along the system are maintained for each designated location. An example of output data from the simulation of the south-bound 405 Freeway with is included. The traffic was assumed to be 2.2 times the ADT.

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[1] R. Evans, The Atomic Nucleus, (McGraw-Hill, 1955)