Multiple-Slot Clear-Path Operational Control

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Introduction

It’s inherent in the Clear-Path concept that NO vehicle will EVER be allowed entry onto the guideways unless it has been positively established that a clear path all the way to the destination is available and the relevant slots have been reserved for said vehicle’s exclusive use. This means that ALL conflicts have to be solved ’a priori’ and the only method available to achieve this is ’entry delay’. Vehicles will simply have their entry delayed until such time that the necessary vacant slots are available.

The most common criticism leveled at the Clear-Path concept is pertinent to this method of ’entry delay’, since it is often claimed through the application of simple math that the probability of finding the necessary vacant slots will rapidly approach zero as the number of intersections passed en route goes up – especially if the guideways are intensively utilized. This in turn will result in very long and completely impractical delays for users trying to access the system.

I believe this criticism is severely exaggerated and the simple math used to illustrate the shortcomings of Clear-Path control is misleading as it doesn’t take into account the nature of the Clear-Path principle, the actual traffic patterns and other factors besides that. I will therefore dedicate a good deal of this presentation to a more detailed examination of a Single-Slot Clear-Path system and its characteristics. Only after that will I develop the Single-Slot Clear-Path system into a Multiple-Slot Clear-Path system which will eliminate any remaining weaknesses pertinent to Clear-Path Operational Control, thereby describing a simple, flexible and practical operational control principle suitable for a future dual-mode transportation system.

Why Clear-Path Control?

A Clear-Path Operational Control system is inherently very simple. It solves ALL conflicts prior to launching a vehicle onto the guideway; creating a cleared path of reserved time-slots from the vehicle’s origin all the way to its destination. A ’conflict’ is an occurrence where two or more vehicles are destined to occupy the same location on the guideway at the same time. Such an occurrence requires some means to solve the conflict since it would otherwise result in vehicles colliding. A ’time-slot’ (hereafter called slot) is a precise location on the guideway coupled to an equally specific time.

Solving conflicts ’a priori’ by slaving a vehicle to a specific moving slot allows for a control system where individual vehicles run on guideways without any outside intervention. All that is required is a sufficient precise and robust real-time position verification system; enabling vehicles to autonomously navigate with reference to the pre-programmed clear path (stored in on-board computers) laid out before them – having wayside computers monitor everything, ready to intervene if – and only if – some irregularity occurs warranting intervention.

The possibility of running vehicles autonomously, with reference only to a pre-programmed position will in turn cut down on the necessary sensors mounted on vehicles. With little need to orient themselves dynamically according to other vehicles will there be little or no need for fault-prone sensors on the vehicles, thus one source (faulty sensors) of potential mishaps and disastrous accidents has been virtually eliminated. Also, there will be very little data transmitted to and from vehicles in the course of trips, thus data will not get lost (or have to be duplicated to assure redundancy) to quite the same degree as for a control system reacting dynamically to the traffic currents.

A system using Clear-Path Operational Control will also undo the need to occasionally stop and temporarily hold back one or more vehicles due to traffic congestion on the guideways. Even if the need to completely stop a vehicle or two is relatively infrequent will there still be a need for deceleration and acceleration ramps connecting one or more "waiting bays" to the guideways. If these ’waiting bays’ are inserted at junctions will it severely limit the capacity of the lines transferring to and from the main lines if vehicles are completely stopped there. And the transfer lines will of cause have to be long enough to facilitate both the deceleration, the ’waiting bays’ as well as the acceleration back to full line-speed, imposing a further design restriction on a system intended to be incorporated into an urban environment where space is already at a premium.

Given the autonomous operation capability of each vehicle in a Clear-Path control system will there be no reason to arrange them into platoons or packets; in fact, a Clear-Path control system might completely exclude that option. It is therefore possible to do without mechanical coupling devises (a source of breakdowns and subsequent mishaps/accidents), just as there will be no need for vehicles to ’bump’ each other at modest relative speeds in order to form the platoons/packets. Anyone that have tried to ’bump’ into something at even 1-2 mph knows how unpleasant that is, since one is thrown forward when hitting an obstacle – even at that modest speed. Facilitating even lower relative ’platoon/packet forming’ speeds will require a very precise control of the speed during ’impact’ with the preceding vehicle. The Clear-Path principle do away with all these complications by doing away with the need for and desirability of platooning.

Operating vehicles in singletons, outside of platoons/packets, will certainly warrant a lot of data, that will have to be verified, processed, transported and stored. Each trip will ’produce’ a lot of storable data about the route taken, the proceedings of the journey, the time, the price for the user, any irregularities during the journey and possibly many other pieces of information, but unless the proponents of platoons envision that users will be invoiced as a group rather than as individual users will more or less the same data have to be produced, processed and stored regardless.

Anyway, producing, processing and storing data is so inexpensive today that I have trouble seeing the amount of data as a problem to operating vehicles in singletons. I will attempt to illustrate this with a reference to a simple phone call. If I call someone in the same city to deliver a 10 seconds message will my phone company register which number I call from, which number I’m calling, what time of the day I call (so that I get invoiced the correct rate) and for how long. They will also register if my call ’uses’ lines of other phone companies, so they can in turn pay these companies for my use of the lines. All this information will they register and print (except the usage of ’foreign’ lines) on my monthly bill, just to collect slightly less than 2 cents from me! Data have become so inexpensive as to be of no importance in relation to which control principle should be preferred for a possible dual-mode system.

Single-Slot Clear-Path Operational Control

In order to realistically evaluate the viability and utility of a given control system will it be necessary to provide some beforehand premises concerning the net of guideways and the traffic travelling these guideways: I therefore submit we have a uniform network of crossing guideways; each guideway line being uni-directional, meaning that each guideway crossing (junction) provides only ONE alternative to staying on the line, namely a transfer to the other, crossing line. A bi-directional line would present TWO alternatives (i.e. north-going and south-going) and would therefore in the following examples count as two ’junctions’.

Furthermore will I assume that a trip will cross 10 junctions and involve 2 line-transfers on average. This again means that 80% of the traffic at any random junction will continue ’straight ahead’ on the same line while the remaining 20% will transfer to the crossing line. Last will I assume that the guideway network is utilized to 60% of theoretical capacity, which I consider a fairly intensive utilization of a high-capacity system. These premises are obviously somewhat arbitrarily picked given our limited beforehand knowledge, so one could always design other scenarios and test the numbers on these.

I have repeatedly seen the probability of any random slot providing a clear path all the way to a destination expressed by the following simple formula: (1 - u)^n where u is the average utilization of the guideways (i.e. 30% = 0,3) and n is the number of junctions where two guideways merge. Inserting my assumptions from above in the formula we get: (1 – 0,6)^10 = 0,0001 or the equivalent of 0,01% chance that a random slot can provide the clear path we need to commence the trip (the corresponding figure for a 50% utilization rate is 0,04%). Such small probabilities will obviously results in outrageously long ’entry delays’ and would of cause be completely unacceptable.

However, this formula is very simplistic and doesn’t take into account the principle of a Clear-Path Operational Control system where all slots are booked immediately before a journey commences. This means that the actual utilization figure for each junction a vehicle will cross should only include those other vehicles which has already booked their slots PRIOR to when our vehicle makes the reservation. Vehicles booking slots AFTER we have booked ours will not ’monopolize’ the guideways seen from our point of view since we already have our slots secured, but these later vehicles will still be part of the final utilization percentage.

It should be obvious that the very first junction encountered on a trip will have been ’monopolized’ to a much higher degree than the very last junction encountered, simply because almost all other vehicles have already booked slots for our first junction while very few vehicles have booked slots at our last junction by the time we book. Other vehicles’ ’monopolization’ of the junctions will, considering my previously mentioned premises, drop off with a constant (equal to 10% of the utilization rate in this example). Therefore, instead of a probability of

(1 – 0,6)^10 we now get:

(1 – 0,60) (1 – 0,54) (1 – 0,48) (1 – 0,42) (1- 0,36) (1 – 0,30) (1 – 0,24) (1 – 0,18)

(1 – 0,12) (1 – 0,06)

which is 0,0128 or 1,28% (the corresponding figure for 50% utilization rate is 3,27%). In other words, taking into account the simple fact that junctions are only ’monopolized’ by vehicles booking slots at an earlier time than the vehicle in question has in this particular example improved the probability of finding a clear path roughly hundredfold compared to the simple formula often used by ’Clear-Path Control’ adversaries.

Unfortunately, 1,28% probability is still a low figure, so I’ll turn my attention to another ’unrealistic’ part of the criticism put forth: Namely the notion that the overall utilization rate of a guideway network is universally applicable to the merge-points (junctions) in question when we do our probability calculations. If we consider two lines, A and B, crossing each other, both utilized at 60% of full capacity, with 80% of that traffic continuing on the same line and 20% transferring to the other line it follows that line A and B will only be utilized 48% (60% x 80%) at the junction and the two transfer lines from A to B and B to A will only be utilized 12% (60% x 20%). Taking the continuing line and the transferring line we see that they are only utilized (48 + 12)/2 = 30% on average. In other words, the transfer line will temporarily ’steal’ traffic from the continuing line, thereby lowering (to half) the average utilization rate at junctions. Using a 30% average utilization rate in the previous formulas would certainly lead to more ’benign’ results than when using a 60% figure.

However, it’s even more complicated than that. We also have to take into consideration that an entering vehicle searching for vacant slots will not ’compete’ for slots with those vehicles taking the same course. If we therefore consider a vehicle that has already found a vacant slot on a guideway utilized at 60% and the vehicle is scheduled to continue ’straight ahead’ at the first junction encountered, it follows that the 80% of traffic (occupying 48% of all slots) going ’straight’ will not interfere with our search for a vacant slot after the junction. Our only ’competitors’ are the 20% of traffic (occupying 12% of all slots on the transfer line) transferring from the other line onto our line. We also know that these 12% of slots will merge with the 52% of slots that are vacant on our line. This means that our chance of continuing ’straight ahead’ (= finding a vacant slot after the junction, but on the same line) at the first junction encountered will be (52 – 12)/52 = 0,7692 or around 77%. Using the same method will our chance of a successful transfer to the other line be (88 – 48)/88 = 0,4545 or roughly 45%.

Applying these figures to the previously described example where ’monopolization’ of the guideways will drop off over distance and assuming the 2 line-transfers will occur at junctions nos. 1 and 2 (which is an absolute worst case scenario) we get the following ’neat’ little computation:

(88 – 48)/88 x (89,2 – 43,2)/89,2 x (61,6 – 9,6)/61,6 x (66,4 – 8,4)/66,4 x

(71,2 – 7,2)/71,2 x (76 – 6)/76 x (80,8 – 4,8)/80,8 x (85,6 – 3,6)/85,6 x

(90,4 – 2,4)/90,4 x (95,2 – 1,2)/95,2

which is 0,1239 or a respectable 12,39% (the corresponding figure for 50% utilization rate is 20,34%). This figure is conditioned on all the premises previously listed and is valid for a trip encountering 10 junctions, so we still need to take into account the problem of finding a vacant slot when entering any random guideway line in the first place. This probability is a simple 40% so the final figure will be 0,1239 x 0,4 x 100 = 4,96%. Translating this into ’entry delay’ it means that vehicles will on average wait around 20 slots or 10 seconds at 2 slots per second before they can commence their trip. It also means that 95% of all users will have to wait less than 60 slots – or less than 30 seconds at 2 slots per second. 30 seconds is a shorter time than most traffic lights let us wait for the green light.

All this is not an attempt to argue that a Single-Slot Clear-Path Operational Control system is a viable choice. Not at all! Our probability figures and therefore also the ’entry delay’ would suffer, if the utilization rate used in these examples were further increased or if the average trip were crossing a higher number of junctions. Indeed, it’s unlikely a 60% utilization rate can even be achieved in the above described system given my assumption that all trips are of equal length over 10 junctions. Accommodating more demanding scenarios would require a more flexible system; one that has other options than mere ’entry delay’ as a means to solve conflicts. To that end have I devised the Multiple-Slot Clear-Path Operational Control concept.

Multiple-Slot Clear-Path Operational Control

’Multiple’ means ’more than one’, so in principle it could indicate 2 or 200 alternative slots, except 200 would probably be impractical and 2 doesn’t really offer enough advantages to be worth the trouble. I did some primitive calculations based on a ’5 alternative slots’ Multiple-Slot system as a starting point and it turned out to be such a fortunate choice that I’m going to use only that in the following presentation. Five alternative slots means that a vehicle can either remain in the same slot, jump one or two slots ’forward’ – or drop one or two slots ’back’. This is done by increasing or decreasing speed slightly just long enough to effect the shift from one slot to another. In principle can slot shifts – or ’slot-jumping’ – be done everywhere on the guideway at any time, but for simplicity I envision that it will only be allowed at junctions. Nevertheless, even with this restriction will it be evident that 5 alternative slots to choose from will improve the chance of finding at least one slot that is vacant.

However, the slot we would need to jump to is unfortunately often occupied or blocked by a preceding/following vehicle thereby somewhat reducing our options. A vehicle just in front of our vehicle will prevent any jumps ’forward’ and a vehicle right behind us will prevent any ’falling back’. Our final number of alternative slots will depend on the utilization rate and the exact position of other vehicles relative to ours. The bottom line is that a 12% utilization rate (the transfer line from the examples above) will allow for 4,31 alternative slots on average and a 48% utilization rate (the continuing line from the examples above) will allow for 2,58 alternative slots on average. Each specific utilization rate yields a different number of slots (which I manually calculated the hard and painful way for all 10 junctions in the following example). A utilization rate of 100% will allow only one slot (the one we already occupy) while a utilization rate of 0% always allows 5 alternative slots.

Returning to the same premises as used in my presentation of the Single-Slot system above we can deduce that the chance of successfully finding a vacant slot at a merge will be 1 – u^s where u is the relative utilization rate (relative to the number of vacant slots on our line) of the line merging with ours and s is the number of alternative slots available to us (which depends on the utilization rate of our line). Again calculating in the diminishing ’monopolization’ of the guideways by other vehicles over the course of a trip we get the following comprehensive computation:

1-(48/88)^4,31 x 1-(43,2/89,2)^4,38 x 1-(9,6/61,6)^2,99 x 1-(8,4/66,4)^3,21 x

1-(7,2/71,2)^3,44 x 1-(6/76)^3,68 x 1-(4,8/80,8)^3,93 x 1-(3,6/85,6)^4,18 x

1-(2,4/90,4)^4,75 x 1-(1,2/95,2)^4,72

which is 0,8829 or 88,29% chance that any random slot will provide a clear path over 10 junctions (including worst case line transfers at junctions nos. 1 and 2), possibly involving one or more ’slot jumps’. Notice that from the seventh term and beyond of the above computation are the results always quasi-ONE (greater than 0,9999) and therefore largely irrelevant for the final probability figure.

The above probability coupled to the initial 40% chance of getting on the guideway in the first place gives 0,8829 x 0,4 x 100 = 35,32% chance that a random slot will provide a clear path. This in turn translates into an average ’entry delay’ of less than 3 slots while 95% of vehicles will find a usable slot within the first 7 slots. With 2 or more slots per second and adding a few seconds needed for the systems’ control computers to search for and book the necessary slots will the average user experience a wait on the order of 3–4 seconds at 60% overall utilization of the guideway network, while only 5% of users will experience waits (slightly) longer than 5–6 seconds. That, in my opinion, is an entirely acceptable wait and I doubt any other Operational Control principle can undercut these figures by much.

But maybe I was just lucky when I choose the 60% utility rate as a basic premise for my calculations? Could it be that 60% is the maximum achievable, since a higher utilization rate works heavily against the system given the math in the example above? I ran the numbers for a 75% utilization rate just in order to test if 60% is the practical limit. A 75% utilization rate yields a 63,75% chance that a random slot will provide a clear path over 10 junctions including two transfers and when the initial entry on the guideway is included will it be 0,6375 x 0,25 x 100 = 15,94%. This is less than half of the figure for a 60% utilization rate, but translated into ’entry delay’ is it still less than 7 slots on average and 95% of users will find a usable slot within the first 17 slots. Using the above premises; that’s 5–6 seconds wait on average and only 5% will have to wait (slightly) longer than 10–12 seconds, so it seems a Clear-Path system operating with a 75% utilization rate is also a viable option.

One final issue should be briefly mentioned here: In an urban setting with a comprehensive network of guideways criss-crossing the city will there possibly and likely be more than one route connecting A and B; which will further increase the probability of finding a clear path very quickly – providing the system’s control computers can search for all routes of more or less equal length simultaneously.

Concluding words

While I make no pretence as to have presented a complete or even a fulfilling case for the Clear-Path principle, I hope to at least have planted the seed of doubt among the many that previously have discarded a Clear-Path Operational Control system as near-impossible to implement and overly limited in utility even if it could be implemented. I see the Clear-Path principle as the better, more capable, simpler and safer option; and while there are still issues to deal with, these are not of bigger magnitude than what can be said about any other operational control principle contending for a future high-capacity dual-mode transportation system.

**Last modified: February 11, 2003**