scholarly journals A unified approach for large queue asymptotics in a heterogeneous multiserver queue

2017 ◽  
Vol 49 (1) ◽  
pp. 182-220 ◽  
Author(s):  
Masakiyo Miyazawa
Keyword(s):  
Queue Length ◽  
Heavy Traffic ◽  
Weak Limit ◽  
Tail Asymptotics ◽  
Heterogeneous Servers ◽  
Unified Approach ◽  
Good Test ◽  
Traffic Condition ◽  
Test Function ◽  
Heavy Traffic Approximations

Abstract We are interested in a large queue in a GI/G/k queue with heterogeneous servers. For this, we consider tail asymptotics and weak limit approximations for the stationary distribution of its queue length process in continuous time under a stability condition. Here, two weak limit approximations are considered. One is when the variances of the interarrival and/or service times are bounded, and the other is when they become large. Both require a heavy-traffic condition. Tail asymptotics and heavy-traffic approximations have been separately studied in the literature. We develop a unified approach based on a martingale produced by a good test function for a Markov process to answer both problems.

Stochastic bounds for heterogeneous-server queues with Erlang service times

1974 ◽  
Vol 11 (04) ◽  
pp. 785-796 ◽  
Author(s):  
Oliver S. Yu
Keyword(s):  
Steady State ◽  
Queue Length ◽  
Heavy Traffic ◽  
Waiting Times ◽  
Single Server ◽  
Shape Parameters ◽  
Server Queue ◽  
Service Times ◽  
Heavy Traffic Approximations ◽  
Multi Server

This paper establishes stochastic bounds for the phasal departure times of a heterogeneous-server queue with a recurrent input and Erlang service times. The multi-server queue is bounded by a simple GI/E/1 queue. When the shape parameters of the Erlang service-time distributions of different servers are the same, these relations yield two-sided bounds for customer waiting times and the queue length, which can in turn be used with known results for single-server queues to obtain characterizations of steady-state distributions and heavy-traffic approximations.

Stochastic bounds for heterogeneous-server queues with Erlang service times

1974 ◽  
Vol 11 (4) ◽  
pp. 785-796 ◽  
Author(s):  
Oliver S. Yu
Keyword(s):  
Steady State ◽  
Queue Length ◽  
Heavy Traffic ◽  
Waiting Times ◽  
Single Server ◽  
Shape Parameters ◽  
Server Queue ◽  
Service Times ◽  
Heavy Traffic Approximations ◽  
Multi Server

This paper establishes stochastic bounds for the phasal departure times of a heterogeneous-server queue with a recurrent input and Erlang service times. The multi-server queue is bounded by a simple GI/E/1 queue. When the shape parameters of the Erlang service-time distributions of different servers are the same, these relations yield two-sided bounds for customer waiting times and the queue length, which can in turn be used with known results for single-server queues to obtain characterizations of steady-state distributions and heavy-traffic approximations.

Heavy Traffic Approximations of a Queue with Varying Service Rates and General Arrivals

2012 ◽  
Vol 28 (1) ◽  
pp. 63-108 ◽  
Author(s):  
Robert Buche ◽  
Arka P. Ghosh ◽  
Vladas Pipiras
Keyword(s):  
Heavy Traffic ◽  
Service Rates ◽  
Heavy Traffic Approximations

Heavy traffic queue length scaling in switches with reconfiguration delay

2021 ◽  
Author(s):  
Chang-Heng Wang ◽  
Siva Theja Maguluri ◽  
Tara Javidi
Keyword(s):  
Queue Length ◽  
Heavy Traffic ◽  
Length Scaling

scholarly journals Pathwise optimality of the exponential scheduling rule for wireless channels

2004 ◽  
Vol 36 (04) ◽  
pp. 1021-1045 ◽  
Author(s):  
Sanjay Shakkottai ◽  
R. Srikant ◽  
Alexander L. Stolyar
Keyword(s):  
Queue Length ◽  
Wireless Channels ◽  
Unique Feature ◽  
Heavy Traffic ◽  
Wireless Channel ◽  
Service Rate ◽  
Resource Pooling ◽  
Scheduling Policy ◽  
Multiple Data ◽  
Heavy Traffic Limit

We consider the problem of scheduling the transmissions of multiple data users (flows) sharing the same wireless channel (server). The unique feature of this problem is the fact that the capacity (service rate) of the channel varies randomly with time and asynchronously for different users. We study a scheduling policy called the exponential scheduling rule, which was introduced in an earlier paper. Given a system withNusers, and any set of positive numbers {an},n= 1, 2,…,N, we show that in a heavy-traffic limit, under a nonrestrictive ‘complete resource pooling’ condition, this algorithm has the property that, for each timet, it (asymptotically) minimizes maxnanq̃n(t), whereq̃n(t) is the queue length of usernin the heavy-traffic regime.

HEAVY TRAFFIC APPROXIMATIONS FOR A SYSTEM OF INFINITE SERVERS WITH LOAD BALANCING

1999 ◽  
Vol 13 (3) ◽  
pp. 251-273 ◽  
Author(s):  
Philip J. Fleming ◽  
Burton Simon
Keyword(s):  
State Space ◽  
Heavy Traffic ◽  
Joint Probability ◽  
Mass Function ◽  
Accurate Estimate ◽  
The State ◽  
Probability Mass Function ◽  
Traffic Loads ◽  
Wide Range ◽  
Heavy Traffic Approximations

We consider an exponential queueing system with multiple stations, each of which has an infinite number of servers and a dedicated arrival stream of jobs. In addition, there is an arrival stream of jobs that choose a station based on the state of the system. In this paper we describe two heavy traffic approximations for the stationary joint probability mass function of the number of busy servers at each station. One of the approximations involves state-space collapse and is accurate for large traffic loads. The state-space in the second approximation does not collapse. It provides an accurate estimate of the stationary behavior of the system over a wide range of traffic loads.

scholarly journals Steady-state analysis of a multiserver queue in the Halfin-Whitt regime

2008 ◽  
Vol 40 (2) ◽  
pp. 548-577 ◽  
Author(s):  
David Gamarnik ◽  
Petar Momčilović
Keyword(s):  
Queue Length ◽  
Heavy Traffic ◽  
Length Distribution ◽  
Moment Generating Function ◽  
Compact Representation ◽  
Single Server ◽  
Service Time Distribution ◽  
Spare Capacity ◽  
Queue Length Distribution ◽  
Stationary Queue Length

We consider a multiserver queue in the Halfin-Whitt regime: as the number of serversngrows without a bound, the utilization approaches 1 from below at the rateAssuming that the service time distribution is lattice valued with a finite support, we characterize the limiting scaled stationary queue length distribution in terms of the stationary distribution of an explicitly constructed Markov chain. Furthermore, we obtain an explicit expression for the critical exponent for the moment generating function of a limiting stationary queue length. This exponent has a compact representation in terms of three parameters: the amount of spare capacity and the coefficients of variation of interarrival and service times. Interestingly, it matches an analogous exponent corresponding to a single-server queue in the conventional heavy-traffic regime.

Organization Optimization of Traffic Engineering in Urban CBD

2013 ◽  
Vol 579-580 ◽  
pp. 890-893
Author(s):  
Mei Mei Huang ◽  
Qing Yang ◽  
Shang Lin Xiao
Keyword(s):  
Traffic Flow ◽  
Traffic Engineering ◽  
Queue Length ◽  
Motor Vehicle ◽  
Heavy Traffic ◽  
Optimization Methods ◽  
Difficult Problem ◽  
Signal Timing ◽  
Total Capacity ◽  
Business District

Orderly organization of traffic engineering in the urban CBD (Center Business District) is a difficult problem, with crowding people flow, heavy traffic flow and complex surrounding situation. This paper set CBD along Xinhua Street in Jinhua city center as an example, focused on the organization optimization process of traffic engineering in CBD. Through the survey on traffic engineering status of sections and intersections, it analyzed road congestion characteristics and intersection signal timing with Vissim software emulation, proposed traffic optimization methods as road channelization, intersection signal timing adjustment of Xinhua-Liberation Road. In Xinhua Street section, it can effectively canalized traffic flow by broadening 2 two-way lanes, adding four pedestrian crossing refuges, and separating Motor vehicle and non-motor vehicle separation barrier. It took queue length, number of stops, delay time three indicators as the objective function with the application of Synchro software adjusting the intersection signal timing. As a result, the total queue length could be reduced from 708.5m to 586.6m and total capacity from 2041 pcu/ h to 2838 pcu/ h.

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