Defining and Measuring Service Availability for Complex Transportation Networks

Transcription

1 Journal of Advanced Transportation, Vol. 32, No. 1, pp Defining and Measuring Service Availability for Complex Transportation Networks Charles P. Elms Service Availability of a transportation system is a measure of a performance that has been generally defined according to the reliability and maintainability terms of mean-time-before-failure and mean-time to-restore, as borrowed from the aerospaceldefense industry. While such definitions correctly describe the availability of a system and its equipment to function they do not directly measure the percent of designed and scheduled service available for passenger use. For the more complex transportation systems having multiple tracks and routes, fleets of vehicles, more than two stations and more than one mode of service there are needs for definitions that account for isolated failures that partially interrupt or delay service. Successful definitions of service availability have been based on data that is easily and directly entered in the operating log or automatically collected by Automatic Train Supervision (ATS) system and reports generated by software. The following paper fmt defines measures of service availability in current use and analyzes exact and approximation methods for data collection and computation. Second, the paper postulates and explores classical and new definitions of service availability applicable for complex networks such as Personal Rapid Transit (PRT). Insight is provided for choosing a suitable definition based on the type of transportation network. Introduction Service Availability is a measure of a transportation system s performance to deliver services when demanded. Classical definitions have been expressions such as the ratio of MTBF to the quantity (MTBF+MITR), where MTBF is the mean time between failures of service and MlTR is the mean time to restore service, or the ratio of the quantity Scheduled Service Time - Downtime to Scheduled Service Time. While such simplistic formulae are applicable to simple singletrack shuttle automated people movers, more complex systems having Charles P. Elms is Senior Principal at Lea+Elliott, Inc. in Chantilly, Virgina, USA. Received November 1996; Acceptad October 1997.

2 76 Charles P. Elms multiple tracks, routes, switches, stations and vehicles require definitionsthat account for isolated failures that partially interrupt service and/or cause delays. Such has been successfully applied for dual-lane loops, pinched and collapsed loops, and simple networks having fleets of vehicles operating from a number of stations on different routes. These approaches have utilized a more complex formula that is a weighed summation of route or line availability, whereby the availability of a route or line is the product of System Mode Availability, Fleet Availability and Station Platform Availability defined in terms of failure events and downtime or delay time. These data are easily and directly collected and availability calculated manually, or automatically by the Automatic Train Supervision (ATS) system and reports generated by software. The following paper first defines measures of service availability in current use and examines methods of calculating exactly or approximating availability, with the accuracy of such approximations quantified. Second, the paper postulates definitions of service availability applicable for complex networks such as Personal Rapid Transit (PRT). Two definitions are made, their merits discussed and insight provided on how to specify numerical values of availability from a service point of view and consistent with a system s design. The classical approach indirectly expresses the ability of the system to satisfy demand for service by a formula that measures the MTBF and MITR of each part of the system upon which fulfillment of service demand is dependent. This approach provides a statistical availability measure of the percent of the system that is operational. Another approach would be to define service availability for a given time period as the product of the probability to successfully make a trip and percent on-time performance if a trip is made. The probability to successfully make a trip could be measured as the ratio of total trips completed to total trips demanded. On-time performance could be measured by the ratio of the theoretical trip time to the actual trip time for all trips carried. Theoretical trip times may be fixed, or variable as a function of demand and traffic conditions, but under the assumption that all equipment and infrastructure is operational and there are no failures. Actual trip time could be measured by accumulating the actual trip times for all trips completed. For variable theoretical trip times the approach factors out the effect of trip delays caused by saturation due to demand. In the case of fixed theoretical trip time the measure includes delays due to demand and traffic congestion as well as failures; therefore, it would better reflect the service performance experienced by passengers. Either case also

3 DeJning and Measuring Service Availability for Complex provides built-in forgiveness for failures where the system can reroute vehicles around failures to meet designed performance. Definitions of Availability in Current Use Classically availability (A) has been defined in terms of reliability (MTBF) and maintainability (MTTR) as follows: A = MTBF = 1 - MTBF + MTTR MTTR = I-u, (1) MTBF + MTTR or A = Scheduled Time - Downtime = 1 - Downtime = 1 - U; (2) Scheduled Time Scheduled Time where unavailability (U) is defined as u = MTTR = Downtime. (3) MTBF + MTTR Scheduled Time The definition of (1) is not calculable directly from operating data; therefore, the definition of (2) is more prevalent, being that it can be calculated from the downtime attributed to failure events that occur within a scheduled period of service. Shuttle Systems Equations (2) and (3) are directly applicable to simple shuttle systems where the failure of the system, track, train or a station causes the total system to be out of service - i.e., the system is either "on" or "off". However, as complexity is introduced, and a failure disrupts service to only a portion of the system, a more complex definition is required. For example, a dual-track shuttle system having a separate train on each track, a single-track shuttle system having more than two stations, and a single-track shuttle with two trains and a center bypass can have failures where partial or degraded services may be continued dependent upon the kind and location of the failure. In each of these cases credit can be given for continuing degraded operations. For the dual-track shuttle, failure of one track would render 50 percent of system operable which is easily accounted for by defining system availability as the weighed sum of the availabilities of each track (A = A,T,/T +A2T2/T),

4 78 Charles P. Elms where T, and T2 are the respective time periods each track is scheduled to be operated and T = T, + T2. See also the general formula of (8) below. In the case of a single-track shuttle with three stations, failure of the train or the middle station disrupts total service, but failure of either end station or the track from the middle to either end station leaves one-third of system services operable. This could be accounted by counting any downtime for the whole system, the train or the middle station at full value and any downtime that interrupts service to the end stations at twothirds value. For any system in which failures can result in a degraded operation, credit for the degraded services could be provided by a weighed sum of normal and degraded service availabilities as follows: where Tn = Time period in normal service including downtime, T, = Time period in degraded service, To = Time no services are operated T = Total time period T, + Td, A, = Availability of normal service during Tn, A, = Availability of degraded service during Td, and K = Fraction of normal services rendered by degraded operation. Applying (2) and (4) to the single-track, three station shuttle results in where D, = Downtime that totally interrupts service and D, = Downtime that only interrupts service to either end station. Similarly the availability for the single-track shuttle with two trains and a center bypass would be

5 Deflning and Measuring Service Availability for Complex where D, = Downtime of a station or a stopped train anywhere except on its bypass track and D, = Downtime when a train is stopped on its bypass track. Lines, Loops, and Simple Networks Equations (2) and (3) are insufficient to done define availability for lines and loops having fleets of operating vehicles and more than two stations and networks of independent lines, sections or routes. For these cases the availability of an independent section of a system is defined as the product of the availabilities of the parts upon which the section is dependent. Ai = n A,, where x denotes each dependent part. (7) L= I The availability of the system is the weighed sum of the availabilities of the independent sections. A = m i = l h,ai, where hi is the fraction of the total services of the system scheduled to be rendered during time period T by section i, and the sum of all hi is equal to one. Line or Loop Availability Applying (7) to a line or loop one defines where Am is the availability of the scheduled operating mode (i.e., the line or loop is not totally down or in some permissible degraded mode); A, is the availability of the fleet required for the scheduled mode; and kp is the availability of all stations required for the scheduled mode. Applying (2) and (3) to (9)

6 80 Charles P. E h A, = 1-U, = l-d,/t, where D, = All countable downtime (e.g., hours) of scheduled mode, D, = All countable downtime of each vehicle (e.g., vehiclehours), v = The number of vehicles required for the scheduled mode, D, = All countable downtime of each station platform or platform doors in the case of coordinated vehicle/platform doors (e.g., platform-hours or door-hours), and p = The number of station platforms or platform doors required for the scheduled mode. Countable downtime is that which has not been specified as exempted; such as interruption or delays caused by passengers, unauthorized persons, animals, intrusion by inanimate objects, loss of utility services, periods that specified climatic limits are exceeded, permissible delays to regulate train operations, and downtime less than a specified threshold (e.g., 60 seconds). Accounting for Demaded Line or Loop Service Where a line or loop includes features for continuing service in a degraded mode for certain failures (e.g., single tracking around a blockage, truncated operations at intermediate turnbacks, separate shuttle operations between certain stations, and combinations of such operations) credit can be given for operation of degraded services by applying (4) to (9). where T, = Time period in the scheduled operating mode including downtimes associated with transitions to degraded service,

7 DeJning and Measuring Service Availability for Complex Tdj = Time period in degraded mode j, T = Total time period T, + CTdj, $ = Fraction of scheduled mode services operated by degraded mode j, Admj = Availability of the degraded mode j defined like (lo), Advj = Availability of the fleet required for degraded mode j defined like (1 l), and Adpj = Availability of station platforms or doors for degraded mode j defined like (12). SimDle Networks A simple network may be composed of a number of different lines or loops over which specified routes are operated. Applying the definition of (9) to (8) results in i = l i= I Route designation, Fraction of the total system services (e.g., passengerplace-miles) scheduled to be operated on route i during time period T, Mode availability for route i defined like (lo), Fleet availability for route i defined like (1 I), and Station platforms availability for route i defined like (12). Degraded services can be accounted for by substituting (13) for A, in (14). a where j designates a route i degraded mode, the definitions of the terms for (14) apply and where

8 82 Charles P. Elms Tfi = Time period in the scheduled operating mode for route i including downtimes associated with transitions to degraded service, Tdj = Time period in degraded mode j for route i, T = Total time period Tfi + CTdj, kj = Fraction of scheduled mode services operated by route i in degraded mode j, Admj = Availability of the route i degraded mode j defined like (lo), Advj = Availability of the fleet required for route i degraded mode j defined like (ll), and Adpj = Availability of station platforms for route i degraded mode j defined like (12). Practical Implementation of Availability Definitions The complexity of definitions provided by equations (13), (14) and (15) call for simplification and ease of collecting data. Therefore, it is suggested that practical definitions are those which calculate availability directly from operating data that is normally recorded in the operating logs, either manually, automatically or both. Operating data are basically events, counts and time; such as failures, service interruptions, service restoration, actual and required operating fleet in operation, actual and required numbers of station platforms (or doors) in operation, and time durations for events and counts. Equations (2) and (3) can be used to calculate subordinate Ai, in terms of Ui, for calculating system availability from (13), (14) or (15). To better understand the process, and the potential for approximations, it is useful to express availability in terms of the unavailability of dependent parts. = 1 - U, - U, - Up + Urn U, + Urn Up + U, Up -t U, U, Up (17) Very high system availability is required, generally above 99% for a transportation system to be considered dependable. Therefore, the unavailability terms of (17) should be very small, hence the terms that are the products of unavailability will be even smaller and possibly negligible. In this case the following approximation is suggested.

9 Defining and Measuring Service Availability for Complex APPROX [A, Ay % 1 = 1 - U, - U, - Up = 1 - D, /T - D, /vt- D, /pt (18) Figure 1 shows the error of such an approximation for equal values of U,, U, and Up when varied over a range of 0.01 percent to 5 percent. For approximated availability of 98 percent and above the error is essentially zero and is negligible for 95 to 98 percent. System availability below 98 percent would be unacceptable; therefore, the suggested method of approximation is considered valid for the subordinate availabilities, Ai, in determining total system availability from (13), (14) or (15). Relating Classical Definitions to Availability of Service Classical definitions of availability of the form of (7) and (8) measure the fraction of a system that is fully operational and do not explicitly measure the percent of scheduled service that is available. From a passenger s point of view, service availability may simply be defined as A, = 1 - Delay / Scheduled Travel Time. (19) The definition of (19) cannot ordinarily be used to specify availability because it is not practical to measure delays and scheduled travel time on a per passenger basis. Also, the measure would not compensate for the effect that demand approaching saturation of system capacity would have to reduce measured availability. However, one can relate (19) to the classical definitions having the form of (9) and get insight for quantitatively specifying system availability according to (9), (13), (14), Figure 1. Error in Approximating Availability

10 84 Charles P. Elms and (15). To guaranty a passenger that he/she will not incur delays exceeding what is specified according to the definition of (19) one must assume that the passenger may demand service at any time. Under this assumption all delays incurred by the system would also be passenger delays. Therefore, the definition of (19) may be used to determine a quantitative value for specifying service availability that can be measured by the forms of (7), (8), (9), (13), (14), and (15). For example, a service availability of 99.5 percent will guaranty that not more than 5 percent of passengers demanding service are delayed by more than 10 percent of their scheduled travel time. Defining Availability for PRT Networks Application the Classical DeJinition The definitions of (14) and (15) have applicability where the system can be divided into its independent parts, including predetermined routes which require specific fleets of vehicles, involve a specific set of stations and switches. Such is not practical for PRT systems providing non-stop ondemand personal service between any two station pairs in the network. Each personal trip may involve different stations and a unique route. For even the simplest of PRT networks the number of potential routes becomes numerous and so great for large networks that the approach is impractical. The following applies the classical definition of the form of (7) to the theoretical PRT network of Figure 2. where: A, = availability of the fleet required for service, as measured by (11). A, = availability of stations; taking into account the failures of associated switches, off-line guideway, and platform doors that render boarding and deboarding platform areas unavailable. The definition of (12) is applicable where platforms are redefined as the boarding and deboarding areas.

11 DeJining and Measuring Service Availability for Complex Figure 2. Theoretical PRT Network A, = availability of interchanges between crossing guideways, taking into account all failures that would prevent use of the interchange, including associated switches, connecting guideway exit and entrance ramps and any stalled vehicles inside the interchange. A, = 1 -U, = 1 -D, /ct, (21) where D, c = All downtime for all interchanges, = Total number of interchanges, and T = Total time period analyzed. A, = availability of mainline track links, taking into account failures that prevent a vehicle from operating through the track link; e.g., from point A to point B in Figure 2. A, = 1 -uk = 1 -Dk/ kt, (22)

12 86 Charles P. Elms where D, k = All downtime for all mainline track links and = Total number of mainline track links. Mainline track links are defined as track sections from points "A" to "B", comprising the entire network of mainline guideway, excluding all interchange guideway and off-line guideway. Application of the definition of (20) requires counting the downtime for each of the four categories of system dependent parts and proper allocation of failures to the respective part not available for use. For example, failure of a switch associated with an interchange that renders the interchange and an associated mainline track link unusable would be counted as a failure of the interchange and the associated mainline track link. However, where only the interchange or the mainline track link is usable the failure would be counted only for the unusable part. Similarly, the failure of a switch associated with an off-line station that renders the station and an associated mainline track link unusable would be counted as a failure of the station and the associated mainline track link. However, where the mainline track link is usable the failure would be counted only for the station. Following the similar arguments as those of Section 3 above (20) can be approximated as follows: AF'PROX [&&A,Ak] = 1 -U, -Us-Uc-uk = 1 - DJvT - D,/sT - DJcT - %/kt (23) Equations (20) and (23) are based on classical definitions for reliability/availability of equipment and systems and do not directly measure service performance. In this case measured availability would be independent of external disturbances, such as demand. For example, it would not measure decreased service performance as demand increases toward system capacity saturation. Also, the measure does not provide any fault tolerance to give credit for the ability of a system to reroute vehicles around a failure and still meet specified performance. Because, the measure is independent of such external disturbances it may be considered primarily appropriate for measuring hardwarelsoftware performance and not design goodness. The insight provided in Section 4 above is expected to apply for numerically specifying availability as defined by (20) or (23).

13 Defining and Measuring Service Availability for Complex A Service Based Definition Where it is considered important to express system availability in terms of service performance, or there are problems in reporting and accumulating the detailed data to apply equations (20) or (23), a service based definition is proposed. In this case Service Availability is defined in terms of the probability of successfully making a demanded trip on time as follows: A = BJ, (24) where B is the probability of successfully making a demanded trip and J is a measure of on-time performance for a completed trip. B and J may be measured as follows: B = Total Trips Completed / Total Trips Demanded (25) and J = C Theoretical Trip Time / C Actual Trip Time, for only the trips completed. (26) The Theoretical Trip Time for a specific demanded trip may be either fixed or variable. If fixed, Theoretical Trip Time could be the design trip time that results from the system design, and found from a lookup table containing predetermined trip times for all possible station pairs in the system. In this case the availability measure would not compensate for demand that approaches saturation of system capacity and traffic congestion, such that the resulting lower performance would be reflected as reduced availability. It would also provide a degree of fault tolerance, giving credit for the ability of a system to reroute vehicles around a failure to better meet specified performance. While such a measure would be a true measure of "service" availability it would not directly reflect equipment and system availability except where all parts of the system are operating within design capacity and traffic congestion does not occur. Variable Theoretical Trip Time might be determined from a "real time" simulation of trip time made at the time the trip is demanded under the assumption of no equipment or system failures, current demands for trips, assigned trips and trips underway. In this case, the effects of demand and current traffic would be reflected in the Theoretical Trip

14 aa Charles P. Elms Time. As system capacity becomes saturated both the numerator and denominator of (26) would increase at the same rate. While system availability in terms of hardware and software are more directly measured it would also provide the same fault tolerance for rerouting vehicles around a failure discussed above. Conclusions As a transportation network increases in complexity classical measures of availability must account for all parts of the system network on which availability is dependent, and should include compensation for degraded service during failures. For current automated people movers this may include the scheduled operation mode, scheduled fleet, stations, lines, loops and routes. Current practice measures availability successfully by application of the comprehensive definition of (15). As a means to reduce the complexity of calculating (15), line or loop availability can be approximated accurately by (18). Measures of availability based on classical reliability and maintainability definitions are direct measures of the ability of hardware and software of a system to perform, but are not direct measures of service performance provided to passengers. However, the two can be directly related for numerically specifying availability according to a maximum delay to passengers, not accounting for delays experienced when demand approaches system capacity. While classical definitions can be applied to complex PRT networks, as postulated by (20) or the approximation of (23), such will require recording extensive data and does not compensate for degraded service (i.e., fault tolerance). The alternative service based definition of (24) for variable Theoretical Trip Time may be somewhat less complicated in terms of record keeping and calculation and does provide fault tolerance for rerouting vehicles around failures. Where an expression of service availability is required to directly reflect service performance provided to passengers as function of system reliability, maintainability, system design quality, demand and operating trafftc conditions the service based definition for fixed Theoretical Trip Time is applicable.