158 research outputs found
Join the Shortest Queue with Many Servers. The Heavy-Traffic Asymptotics
We consider queueing systems with n parallel queues under a Join the Shortest Queue (JSQ) policy in the Halfin-Whitt heavy-traffic regime. We use the martingale method to prove that a scaled process counting the number of idle servers and queues of length exactly two weakly converges to a two-dimensional reflected Ornstein-Uhlenbeck process, while processes counting longer queues converge to a deterministic system decaying to zero in constant time. This limiting system is comparable to that of the traditional Halfin-Whitt model, but there are key differences in the queueing behavior of the JSQ model. In particular, only a vanishing fraction of customers will have to wait, but those who do incur a constant order waiting time. Keywords: queueing theory; parallel queues; diffusion model
Load Balancing in the Non-Degenerate Slowdown Regime
We analyse Join-the-Shortest-Queue in a contemporary scaling regime known as
the Non-Degenerate Slowdown regime. Join-the-Shortest-Queue (JSQ) is a
classical load balancing policy for queueing systems with multiple parallel
servers. Parallel server queueing systems are regularly analysed and
dimensioned by diffusion approximations achieved in the Halfin-Whitt scaling
regime. However, when jobs must be dispatched to a server upon arrival, we
advocate the Non-Degenerate Slowdown regime (NDS) to compare different
load-balancing rules.
In this paper we identify novel diffusion approximation and timescale
separation that provides insights into the performance of JSQ. We calculate the
price of irrevocably dispatching jobs to servers and prove this to within 15%
(in the NDS regime) of the rules that may manoeuvre jobs between servers. We
also compare ours results for the JSQ policy with the NDS approximations of
many modern load balancing policies such as Idle-Queue-First and
Power-of--choices policies which act as low information proxies for the JSQ
policy. Our analysis leads us to construct new rules that have identical
performance to JSQ but require less communication overhead than
power-of-2-choices.Comment: Revised journal submission versio
Universality of Load Balancing Schemes on Diffusion Scale
We consider a system of 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 randomly
selected servers otherwise . This load balancing scheme
subsumes the so-called Join-the-Idle Queue (JIQ) policy and the
celebrated Join-the-Shortest Queue (JSQ) policy 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 , implying that
assigning tasks to idle servers is sufficient for diffusion level optimality
Exponential Tail Bounds on Queues: A Confluence of Non-Asymptotic Heavy Traffic and Large Deviations
In general, obtaining the exact steady-state distribution of queue lengths is
not feasible. Therefore, we establish bounds for the tail probabilities of
queue lengths. Specifically, we examine queueing systems under Heavy-Traffic
(HT) conditions and provide exponentially decaying bounds for the probability
, where is the HT parameter denoting how
far the load is from the maximum allowed load. Our bounds are not limited to
asymptotic cases and are applicable even for finite values of , and
they get sharper as . Consequently, we derive non-asymptotic
convergence rates for the tail probabilities. Unlike other approaches such as
moment bounds based on drift arguments and bounds on Wasserstein distance using
Stein's method, our method yields sharper tail bounds. Furthermore, our results
offer bounds on the exponential rate of decay of the tail, given by
for any finite value of .
These can be interpreted as non-asymptotic versions of Large Deviation (LD)
results.
We demonstrate our approach by presenting tail bounds for: (i) a continuous
time Join-the-shortest queue (JSQ) load balancing system, (ii) a discrete time
single-server queue and (iii) an queue. We not only bridge the gap
between classical-HT and LD regimes but also explore the large system HT
regimes for JSQ and systems. In these regimes, both the system size and
the system load increase simultaneously. Our results also close a gap in the
existing literature on the limiting distribution of JSQ in the super-NDS
(a.k.a. super slowdown) regime. This contribution is of an independent
interest. Here, a key ingredient is a more refined characterization of state
space collapse for JSQ system, achieved by using an exponential Lyapunov
function designed to approximate the norm.Comment: 37 pages, 1 figur
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