Analytical approximation of transient joint queue-length distributions of a finite capacity queueing network

Transcription

1 Analytical approximation of transient joint queue-length istributions of a finite capacity queueing network Gunnar Flötterö Carolina Osorio March 29, 2013 Problem statements This work contributes to the moeling of analytical tractable joint istributions of time-epenent finite capacity Markovian queueing network states. The moel is motivate by an interest to analyse network effects in congeste urban roa networks. The evelopment of analytical, probabilistic an time-epenent moels for urban traffic is selom. In urban transportation, as in most fiels, time-epenent (i.e., transient) analysis is most often carrie out by either the use of (i) eterministic analytical moels (such as the funamental kinematic wave moel (Lighthill an Witham; 1955; Richars; 1956)) or (ii) simulation-base moels (for a review, see Barceló (2010)), which when stochastic can provie istributional estimates that go beyon first-orer information (i.e., expectations). There is currently a lack of analytical transient techniques for finite capacity Markovian queueing networks that account for spatial-temporal epenencies, an even more a lack of tractable (i.e., computationally efficient) techniques. This paper proposes a tractable analytical approximation of transient multi-imensional queue-length istributions. In the broaer fiel of transportation (all moes consiere), few analytical probabilistic an time-epenent techniques have been evelope (for a single queue see Heiemann (2001); Peterson et al. (1995b); for networks of queues see Osorio an Flötterö (2012); Osorio et al. (2011); Gupta (2011); Peterson et al. (1995a); Ooni an Roth (1983)). This work contributes to the probabilistic analytical moeling of urban traffic. The propose network moel can be couple with etaile link moels of urban traffic (Osorio an Flötterö; 2012; Osorio et al.; 2011), which are also erive base on queueing network theory an are consistent with the mainstream eterministic traffic moels. The combination of such moels can escribe in a tractable an etaile manner both the within-link an the between-link ynamics, leaing to a etaile probabilistic escription of the spatial-temporal evolution of urban congestion. Methoology Consier a network of queues in an arbitrary topology, consisting of single server queues. External arrivals to a queue arise following an inepenent Poisson process, an service times are inepenently, ientically istribute exponential ranom variables. Routing is probabilistic. The following events are possible in the network: (i) arrivals from outsie the network into a queue, (ii) epartures from a queue out of the network, (iii) transitions from an upstream queue into a ownstream KTH Royal Institute of Technology, Department of Transport Science, Stockholm, Sween, gunnar.floetteroe@abe.kth.se (corresponing author) Massachusetts Institute of Technology (MIT), Department of Civil & Environmental Engineering, Cambrige, MA 02139, USA, osorioc@mit.eu 1

2 queue. The rates at which these events occur may be time-epenent. All queues have finite capacities, enabling the moeling of spill-back effects. Denoting by N = N(t) the ranom vector of all queue states in the network at real-value time t, the ynamics of the joint istribution of N are guie by the following linear system of ifferential equations (Reibman; 1991): t P(N = y) = x t y xp(n = x) (1) where t P is the time erivative of P, x = (x i) an y = (y i ) are both realizations of N, an t y x is the transition rate from state x into state y. This moel becomes computationally intractable for non-trivial networks ue to the exponential increase in the imension of the state space as a function of the number of queues. This work hence proposes a transient ecomposition technique that characterizes the high-imensional joint istributionp(n) in terms of lowerimensional marginals over non-isjoint ( overlapping ) subnetworks, where a given queue belongs to the subnetworks of both its upstream an its ownstream queues. This ecomposition is not expensive to compute an, as proven in the full paper, unique. Let S be a subnetwork an S be the set of all of its ajacent subnetworks. The ynamics of the state N S, S of all queues in S an S only can then be expresse as follows: t P(N S, S = (m, s)) = n q (n,q) (S, S)P(N S, S = (n, q)) (2) where(n, q) an(m, s) are elements of the combine state space ofsan S, with the respective first tuple element belonging tosan the secon element belonging to S, an (n,q)(s, S) is the transition rate from state(n, q) to (m, s) when only accounting for events possible in subnetworksan S. Summing out the queue states s an rearranging terms results in t P(N S = m) = (n,q) (S, S)P(N S, S = (n, q)) s n q = n [ q s (n,q) (S, S)P(N S = q N S = n) ] P(N S = n) (3) where N S an N S are the ranom vectors of queue states only in subnetworks S an S, respectively. It is noteworthy that the term in brackets functions like a state-epenent transition rate from subnetwork state n to subnetwork state m. This so far exact expression is the basis for the propose moel. The full article evelops in etail a tractable approximation of these state-epenent transition rates. This approximation only requires computations base on subnetwork states, resulting in a moel specification where the ynamics of all overlapping subnetworks are jointly etermine, with each subnetwork only evaluating the instantaneous states of its neighbouring subnetworks. Further, the article proves that if any two subnetwork istributions have ientical marginals for a common set of queues at some point in time, this marginal will also be ientical at all other points in time. This constitutes a key feature of the propose ecomposition approach: It maintains mutually consistent overlapping subnetwork istributions without any nee to introuce supplementary istributional ajustments or constraints. Illustrative results Consier the network shown in Figure 1, consisting of eight single-lane roas. It is constructe in the same spirit as the network presente by Corthout et al. (2012). Two inflows compete for the network capacity, 2

3 Figure 1: Conflicting network inflows Figure 2: Multiple solutions one entering at the south-western link (blue) an one entering at the north-eastern link (orange). The flows split at the intersections proportionally to the withs of the respective arrows. At the merge points, straight flows have priority over flows entering from the sie. A eterministic version of this moel has two solutions, as shown in Figure 2. In the first (top) solution, the blue flow ominates the south-eastern merge, such that the orange flow spills back an cannot enter the north-western merge, where the blue flow can hence enter unhinere. Symmetrically, in the secon (bottom) solution the orange flow ominates the north-western merge. The blue flow hence spills back, without entering the south-eastern merge, where the orange flow can hence pass unhinere. In the presence of stochasticity, this system may oscillate between these two solutions, with the perio of these oscillations epening on the network parameters. Inee, it is possible to construct arbitrarily long perios of these oscillations, making a simulation-base analysis of this system extremely cumbersome. To give an example, Figure 3 shows the occupancy of the mile link in the lower roa stretch, going from west to east, over simulation time. It fluctuates between being almost completely full (containing 9 or 10 vehicles, meaning that the blue flow spills back) an a more istribute state corresponing to maximum flow throughput (meaning that the blue flow has taken over). Figure 4 shows an estimation of the time-epenent expecte occupancy on the same link as compute by the propose analytical moel. The moel captures two important effects. First, its transient nature allows to ientify an initial overshoot in expecte occupancy, resulting from the fact one starts with an empty network, such that initially the eman can enter an occupy the network unhinere from both sies. Only after a while, spillback effects occur. Secon, it takes more than 300 time steps (5 minutes) to attain stationarity in this system, ue to its oscillatory nature. The full article investigates in greater etail how the propose moel approximates multi-variate network istributions of this type. 3

4 Figure 3: Fluctuations of link occupancy over time References Figure 4: Analytically compute expectation of link occupancy over time Barceló, J. (2010). Funamentals of traffic simulation, Vol. 145 of International Series in Operations Research an Management Science, Springer, New York, USA. Corthout, R., Flötterö, G., Viti, F. an Tampere, C. (2012). Non-unique flows in macroscopic first-orer intersection moels, Transportation Research Part B 46(3): Gupta, S. (2011). A framework to span airport elay estimates using transient queuing moels, Technical report, Massachusetts Institute of Technology. Heiemann, D. (2001). A queueing theory moel of nonstationary traffic flow, Transportation Science 35(4): Lighthill, M. an Witham, J. (1955). On kinematic waves II. a theory of traffic flow on long crowe roas, Proceeings of the Royal Society A 229: Ooni, A. R. an Roth, E. (1983). An empirical investigation og the transient behavior of stationary queueing systems, Operations Research 31(3): Osorio, C. an Flötterö, G. (2012). Capturing epenency among link bounaries in a stochastic network loaing moel, International Symposium on Dynamic Traffic Assignment (DTA), Martha s Vineyar, USA. Osorio, C., Flötterö, G. an Bierlaire, M. (2011). Dynamic network loaing: a stochastic ifferentiable moel that erives link state istributions, Transportation Research Part B 45(9):

5 Peterson, M. D., Bertsimas, D. J. an Ooni, A. R. (1995a). Decomposition algorithms for analyzing transient phenomena in multiclass queueing networks in air transportation, Operations Research 43(6): Peterson, M. D., Bertsimas, D. J. an Ooni, A. R. (1995b). Moels an algorithms for transient queueing congestion at airports, Management Science 41(8): Reibman, A. (1991). A splitting technique for Markov chain transient solution, in W. J. Stewart (e.), Numerical solution of Markov chains, Marcel Dekker, Inc, New York, USA, chapter 19, pp Richars, P. I. (1956). Shock waves on highways, Operations Research 4(1):