Abstract. This paper studies the effect of an overdispersed arrival process on the performance of an infinite-server system. In our setup, a random environment is modeled by drawing an arrival rate Λ from a given distribution every ∆ time units, yielding an i.i.d. sequence of arrival rates Λ 1 , Λ 2 , . . .. Applying a martingale central limit theorem, we obtain a functional central limit theorem for the scaled queue length process. We proceed to large deviations and derive the logarithmic asymptotics of the queue length's tail probabilities. As it turns out, in a rapidly changing environment (i.e., ∆ is small relative to Λ) the overdispersion of the arrival process hardly affects system behavior, whereas in a slowly changing random environment it is fundamentally different; this general finding applies to both the central limit and the large deviations regime. We extend our results to the setting where each arrival creates a job in multiple infinite-server queues.
Load balancing algorithms play a vital role in enhancing performance in data centers and cloud networks. Due to the massive size of these systems, scalability challenges, and especially the communication overhead associated with load balancing mechanisms, have emerged as major concerns. Motivated by these issues, we introduce and analyze a novel class of load balancing schemes where the various servers provide occasional queue updates to guide the load assignment.We show that the proposed schemes strongly outperform JSQ(d ) strategies with comparable communication overhead per job, and can achieve a vanishing waiting time in the many-server limit with just one message per job, just like the popular JIQ scheme. The proposed schemes are particularly geared however towards the sparse feedback regime with less than one message per job, where they outperform corresponding sparsified JIQ versions.We investigate fluid limits for synchronous updates as well as asynchronous exponential update intervals. The fixed point of the fluid limit is identified in the latter case, and used to derive the queue length distribution. We also demonstrate that in the ultra-low feedback regime the mean stationary waiting time tends to a constant in the synchronous case, but grows without bound in the asynchronous case.
Abstract-Load balancing algorithms play a crucial role in delivering robust application performance in data centers and cloud networks. Recently, strong interest has emerged in Jointhe-Idle-Queue (JIQ) algorithms, which rely on tokens issued by idle servers in dispatching tasks and outperform power-of-d policies. Specifically, JIQ strategies involve minimal information exchange, and yet achieve zero blocking and wait in the manyserver limit. The latter property prevails in a multiple-dispatcher scenario when the loads are strictly equal among dispatchers. For various reasons it is not uncommon however for skewed load patterns to occur. We leverage product-form representations and fluid limits to establish that the blocking and wait then no longer vanish, even for arbitrarily low overall load. Remarkably, it is the least-loaded dispatcher that throttles tokens and leaves idle servers stranded, thus acting as bottleneck.Motivated by the above issues, we introduce two enhancements of the ordinary JIQ scheme where tokens are either distributed non-uniformly or occasionally exchanged among the various dispatchers. We prove that these extensions can achieve zero blocking and wait in the many-server limit, for any subcritical overall load and arbitrarily skewed load profiles. Extensive simulation experiments demonstrate that the asymptotic results are highly accurate, even for moderately sized systems.
We consider a system of N parallel queues with unit exponential service rates and a single dispatcher where tasks arrive as a Poisson process of rate λ( N ). When a task arrives, the dispatcher assigns it to a server with the shortest queue among d ( N ) ≤ N randomly selected servers. This load balancing policy is referred to as a power-of- d ( N ) or JSQ( d ( N )) scheme, and subsumes the Join-the-Shortest Queue (JSQ) policy as a crucial special case for d ( N ) = N . We construct a coupling to bound the difference in the queue length processes between the JSQ policy and an arbitrary value of d ( N ). We use the coupling to derive the fluid limit in the regime where λ( N )/ N → λ < 1 and d ( N )→ ∞ as N → ∞, along with the corresponding fixed point. The fluid limit turns out not to depend on the exact growth rate of d ( N ), and in particular coincides with that for the JSQ policy. We further leverage the coupling to establish that the diffusion limit in the regime where ( N --λ( N ))/ √ N → β > 0 and d ( N )/ √ N log N → ∞ as N → ∞ corresponds to that for the JSQ policy. These results indicate that the stochastic optimality of the JSQ policy can be preserved at the fluid-level and diffusion-level while reducing the overhead by nearly a factor O( N ) and O(√ N ), respectively.
We consider a time-slotted queueing model where each time slot can either be an arrival slot, in which new packets arrive, or a departure slot, in which packets are transmitted and hence depart from the queue. The slot scheduling strategy we consider describes periodically, and for a fixed number of time slots, which slots are arrival and departure slots. We consider a static and a dynamic strategy. For both strategies, we obtain expressions for the probability generating function of the steady-state queue length and the packet delay. The model is motivated by cable-access networks, which are often regulated by a request-grant procedure in which actual data transmission is preceded by a reservation procedure. Time slots can then either be used for reservation or for data transmission.
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