Heavy Traffic Limits for Join-the-Shortest-Estimated-Queue Policy Using Delayed Information

2020 ◽  
Author(s):  
Rami Atar ◽  
David Lipshutz
Keyword(s):  
Diffusion Model ◽  
Heavy Traffic ◽  
Partial Observations ◽  
Stochastic Delay ◽  
Parallel Queues ◽  
Traffic Conditions ◽  
Current State ◽  
Queue Lengths ◽  
Shortest Queue ◽  
Stochastic Delay Equation

We consider a load-balancing problem for a network of parallel queues in which information on the state of the queues is subject to a delay. In this setting, adopting a routing policy that performs well when applied to the current state of the queues can perform quite poorly when applied to the delayed state of the queues. Viewing this as a problem of control under partial observations, we propose using an estimate of the current queue lengths as the input to the join-the-shortest-queue policy. For a general class of estimation schemes, under heavy traffic conditions, we prove convergence of the diffusion-scaled process to a solution of a so-called diffusion model, in which an important step toward this goal establishes that the estimated queue lengths undergo state-space collapse. In some cases, our diffusion model is given by a novel stochastic delay equation with reflection, in which the Skorokhod boundary term appears with delay. We illustrate our results with examples of natural estimation schemes, discuss their implementability, and compare their relative performance using simulations.

Minimizing response times and queue lengths in systems of parallel queues

1999 ◽  
Vol 36 (04) ◽  
pp. 1185-1193 ◽  
Author(s):  
Ger Koole ◽  
Panayotis D. Sparaggis ◽  
Don Towsley
Keyword(s):  
Hazard Rate ◽  
Counting Process ◽  
Response Times ◽  
Departure Process ◽  
Parallel Queues ◽  
Queue Lengths ◽  
Service Times ◽  
Shortest Queue ◽  
Cost Functionals ◽  
Generally True

We consider the problem of routeing customers to one of two parallel queues. Arrivals are independent of the state of the system but otherwise arbitrary. Assuming that queues have infinite capacities and the service times form a sequence of i.i.d. random variables with increasing likelihood ratio (ILR) distribution, we prove that the shortest queue (SQ) policy minimizes various cost functionals related to queue lengths and response times. We give a counterexample which shows that this result is not generally true when the service times have increasing hazard rate but are not increasing in the likelihood rate sense. Finally, we show that when capacities are finite the SQ policy stochastically maximizes the departure process and minimizes the loss counting process.

Minimizing response times and queue lengths in systems of parallel queues

1999 ◽  
Vol 36 (4) ◽  
pp. 1185-1193 ◽  
Author(s):  
Ger Koole ◽  
Panayotis D. Sparaggis ◽  
Don Towsley
Keyword(s):  
Hazard Rate ◽  
Counting Process ◽  
Response Times ◽  
Departure Process ◽  
Parallel Queues ◽  
Queue Lengths ◽  
Service Times ◽  
Shortest Queue ◽  
Cost Functionals ◽  
Generally True

We consider the problem of routeing customers to one of two parallel queues. Arrivals are independent of the state of the system but otherwise arbitrary. Assuming that queues have infinite capacities and the service times form a sequence of i.i.d. random variables with increasing likelihood ratio (ILR) distribution, we prove that the shortest queue (SQ) policy minimizes various cost functionals related to queue lengths and response times. We give a counterexample which shows that this result is not generally true when the service times have increasing hazard rate but are not increasing in the likelihood rate sense. Finally, we show that when capacities are finite the SQ policy stochastically maximizes the departure process and minimizes the loss counting process.

scholarly journals Universality of load balancing schemes on the diffusion scale

2016 ◽  
Vol 53 (4) ◽  
pp. 1111-1124 ◽  
Author(s):  
Debankur Mukherjee ◽  
Sem C. Borst ◽  
Johan S. H. van Leeuwaarden ◽  
Philip A. Whiting
Keyword(s):  
Load Balancing ◽  
Poisson Process ◽  
Heavy Traffic ◽  
Diffusion Limit ◽  
Parallel Queues ◽  
Special Cases ◽  
Service Rates ◽  
Join The Shortest Queue ◽  
Shortest Queue ◽  
Diffusion Scale

Abstract We consider a system of N parallel queues with identical exponential service rates and a single dispatcher where tasks arrive as a Poisson process. When a task arrives, the dispatcher always assigns it to an idle server, if there is any, and to a server with the shortest queue among d randomly selected servers otherwise (1≤d≤N). This load balancing scheme subsumes the so-called join-the-idle queue policy (d=1) and the celebrated join-the-shortest queue policy (d=N) as two crucial special cases. We develop a stochastic coupling construction to obtain the diffusion limit of the queue process in the Halfin‒Whitt heavy-traffic regime, and establish that it does not depend on the value of d, implying that assigning tasks to idle servers is sufficient for diffusion level optimality.

Driver Acceptance of Unreliable Route Guidance Information

1994 ◽  
Vol 38 (16) ◽  
pp. 1062-1066 ◽  
Author(s):  
Richard J. Hanowski ◽  
Susan C. Kantowitz ◽  
Barry H. Kantowitz
Keyword(s):  
Real Time ◽  
Heavy Traffic ◽  
Traffic Information ◽  
Route Guidance ◽  
Experimental Conditions ◽  
Traveler Information ◽  
Traveler Information Systems ◽  
Traffic Conditions ◽  
Driver Acceptance ◽  
Real Time Information

Human factors research can be used to design safe and efficient Advanced Traveler Information Systems (ATIS) that are easy to use (Kantowitz, Becker, & Barlow, 1993). This research used the Battelle Route Guidance Simulator (RGS) to examine two important issues related to driver behavior and acceptance of ATIS technology: (1) the effect of route familiarity on ATIS use and acceptance and (2) the level of information accuracy needed for an ATIS to be accepted and considered useful. The RGS included two 486 computers that provided drivers with real-time information and traffic reports. Drivers used a touch screen to select routes on one computer monitor and watched the results of their selection (i.e., real-time video of the traffic) on a second computer monitor. Drivers could use the system to obtain information about the traffic conditions on any link before traversing a route. In this experiment, subjects were exposed to four experimental conditions involving manipulation of the driver's familiarity with the route and the reliability of the traffic information obtained from the RGS (i.e., 100%, 71%, and 43% accuracy). The driver's goal was to reach the destination as quickly as possible by avoiding heavy traffic. The results indicated that drivers were able to benefit from system information when it was reliable, but not when it was unreliable. Trust ratings for the 43% accuracy group were significantly higher at the beginning of the four trials than at the end. Also, drivers were more apt to rely on the ATIS and accept information given in an unfamiliar traffic network versus a familiar one.

STRATEGIC DYNAMIC JOCKEYING BETWEEN TWO PARALLEL QUEUES

2015 ◽  
Vol 30 (1) ◽  
pp. 41-60 ◽  
Author(s):  
Amin Dehghanian ◽  
Jeffrey P. Kharoufeh ◽  
Mohammad Modarres
Keyword(s):  
Poisson Process ◽  
Sojourn Time ◽  
Queueing System ◽  
The Other ◽  
Infinite Time ◽  
Numerical Examples ◽  
Parallel Queues ◽  
Time Horizons ◽  
Holding Cost ◽  
Shortest Queue

Consider a two-station, heterogeneous parallel queueing system in which each station operates as an independent M/M/1 queue with its own infinite-capacity buffer. The input to the system is a Poisson process that splits among the two stations according to a Bernoulli splitting mechanism. However, upon arrival, a strategic customer initially joins one of the queues selectively and decides at subsequent arrival and departure epochs whether to jockey (or switch queues) with the aim of reducing her own sojourn time. There is a holding cost per unit time, and jockeying incurs a fixed non-negative cost while placing the customer at the end of the other queue. We examine individually optimal joining and jockeying policies that minimize the strategic customer's total expected discounted (or undiscounted) costs over finite and infinite time horizons. The main results reveal that, if the strategic customer is in station 1 with ℓ customers in front of her, and q1 and q2 customers in stations 1 and 2, respectively (excluding herself), then the incentive to jockey increases as either ℓ increases or q2 decreases. Numerical examples reveal that it may not be optimal to join, and/or jockey to, the station with the shortest queue or the fastest server.

scholarly journals A diffusion model for two parallel queues with processor sharing: transient behavior and asymptotics

1999 ◽  
Vol 12 (4) ◽  
pp. 311-338 ◽  
Author(s):  
Charles Knessl
Keyword(s):  
Diffusion Approximation ◽  
Queue Length ◽  
Heavy Traffic ◽  
Transient Behavior ◽  
Processor Sharing ◽  
Parallel Queues ◽  
Poisson Arrival ◽  
Heavy Traffic Limit ◽  
Service Rates ◽  
Dependent Probability

We consider two identical, parallel M/M/1 queues. Both queues are fed by a Poisson arrival stream of rate λ and have service rates equal to μ. When both queues are non-empty, the two systems behave independently of each other. However, when one of the queues becomes empty, the corresponding server helps in the other queue. This is called head-of-the-line processor sharing. We study this model in the heavy traffic limit, where ρ=λ/μ→1. We formulate the heavy traffic diffusion approximation and explicitly compute the time-dependent probability of the diffusion approximation to the joint queue length process. We then evaluate the solution asymptotically for large values of space and/or time. This leads to simple expressions that show how the process achieves its stead state and other transient aspects.

Adequacy of TRANSYT-7F and Synchro models along a major arterial in Saudi Arabia

2009 ◽  
Vol 36 (1) ◽  
pp. 95-102 ◽  
Author(s):  
Nedal T. Ratrout ◽  
Maen Abdullatif Abu Olba
Keyword(s):  
Saudi Arabia ◽  
Queue Length ◽  
Optimal Timing ◽  
Signal Timing ◽  
Calibration Process ◽  
Traffic Conditions ◽  
Platoon Dispersion ◽  
Queue Lengths ◽  
The City ◽  
Optimal Signal

The TRANSYT-7F and Synchro models are used in developing optimal timing plans in the city of Al-Khobar, Saudi Arabia. This paper evaluates the adequacy of both TRANSYT-7F and Synchro under local traffic conditions by comparing queue lengths observed along a major arterial in the study area with simulated queues. The models were then calibrated to produce simulated queue lengths which are as close as possible to the observed ones. A clear difference was found between queue lengths estimated by Synchro and TRANSYT-7F. A queue length calibration process was accomplished for TRANSYT-7F by using platoon dispersion factor values of 20 and 35 for through and left-turning traffic, respectively. Synchro calibration was unsatisfactory. The simulated queue lengths could not be calibrated in a meaningful way to resemble the observed queue lengths. Regardless of this, both models produced comparable optimal signal timing plans.

scholarly journals Perimeter Control as an Alternative to Dedicated Bus Lanes: A Case Study

2018 ◽  
Vol 2672 (20) ◽  
pp. 110-120 ◽  
Author(s):  
Nicolas Chiabaut ◽  
Michael Küng ◽  
Monica Menendez ◽  
Ludovic Leclercq
Keyword(s):  
Traffic Management ◽  
Transit System ◽  
Control Approach ◽  
Traffic Conditions ◽  
Car Traffic ◽  
Traffic Performance ◽  
Objective Control ◽  
Queue Lengths ◽  
Urban Arterial

Dedicated bus lanes (DBLs) are a common traffic management strategy in cities as they improve the efficiency of the transit system by preventing buses from getting trapped in traffic jams. Nevertheless, DBLs also have certain disadvantages: they consume space, reduce available capacity for general traffic, and can thus lead to even more congested car traffic situations. It is appealing to find more efficient alternatives that maintain a sufficient network supply for general traffic while guaranteeing high commercial speeds for the bus system. This paper investigates whether perimeter control (gating) could be such an alternative to DBL strategies. This solution aims at controlling the traffic conditions of a given area by monitoring vehicle accumulations and adapting traffic signal parameters to reach the targeted conditions. If free-flow states can be maintained within the zone, then DBLs become superfluous. This hypothesis is examined through a simulation case study with an urban arterial acting as the targeted area. A dual-objective control approach was applied to allow for not only the vehicle accumulation inside the area but the queue lengths at its perimeter, thereby addressing one of the main issues associated with gating schemes. Due to the gating strategy, traffic performance in the arterial, measured through vehicle accumulation plus mean speed and density, improved significantly. Moreover, results showed that bus operations reach almost the same efficiency level when DBLs are replaced by perimeter control. Furthermore, the availability of an additional lane for general traffic in the control case significantly increased the arterial capacity for cars.

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