Sensitivity of mean-field fluctuations in Erlang loss models with randomized routing

In this paper we study a large system of N servers, each with capacity to process at most C simultaneous jobs; an incoming job is routed to a server if it has the lowest occupancy amongst d (out of N) randomly selected servers. A job that is routed to a server with no vacancy is assumed to be blocked and lost. Such randomized policies are referred to JSQ(d) (Join the Shortest Queue out of d) policies. Under the assumption that jobs arrive according to a Poisson process with rate $N\lambda^{(N)}$ where $\lambda^{(N)}=\sigma-\frac{\beta}{\sqrt{N}\,}$ , $\sigma\in\mathbb{R}_+$ and $\beta\in\mathbb{R}$ , we establish functional central limit theorems for the fluctuation process in both the transient and stationary regimes when service time distributions are exponential. In particular, we show that the limit is an Ornstein–Uhlenbeck process whose mean and variance depend on the mean field of the considered model. Using this, we obtain approximations to the blocking probabilities for large N, where we can precisely estimate the accuracy of first-order approximations.

Asymptotic Optimality of Power-of-d Load Balancing in Large-Scale Systems

We consider a system of N identical server pools and a single dispatcher in which tasks with unit-exponential service requirements arrive at rate [Formula: see text]. In order to optimize the experienced performance, the dispatcher aims to evenly distribute the tasks across the various server pools. Specifically, when a task arrives, the dispatcher assigns it to the server pool with the minimum number of tasks among d(N) randomly selected server pools. We construct a stochastic coupling to bound the difference in the system occupancy processes between the join-the-shortest-queue (JSQ) policy and a scheme with an arbitrary value of d(N). We use the coupling to derive the fluid limit in case [Formula: see text] and [Formula: see text] as [Formula: see text] along with the associated fixed point. The fluid limit turns out to be insensitive to the exact growth rate of d(N) and coincides with that for the JSQ policy. We further establish that the diffusion limit corresponds to that for the JSQ policy as well, as long as [Formula: see text], and characterize the common limiting diffusion process. These results indicate that the JSQ optimality can be preserved at the fluid and diffusion levels while reducing the overhead by nearly a factor O(N) and O([Formula: see text]), respectively.

Steady-State Analysis of the Join-the-Shortest-Queue Model in the Halfin–Whitt Regime

This paper studies the steady-state properties of the join-the-shortest-queue model in the Halfin–Whitt regime. We focus on the process tracking the number of idle servers and the number of servers with nonempty buffers. Recently, Eschenfeldt and Gamarnik proved that a scaled version of this process converges, over finite time intervals, to a two-dimensional diffusion limit as the number of servers goes to infinity. In this paper, we prove that the diffusion limit is exponentially ergodic and that the diffusion scaled sequence of the steady-state number of idle servers and nonempty buffers is tight. Combined with the process-level convergence proved by Eschenfeldt and Gamarnik, our results imply convergence of steady-state distributions. The methodology used is the generator expansion framework based on Stein’s method, also referred to as the drift-based fluid limit Lyapunov function approach in Stolyar. One technical contribution to the framework is to show how it can be used as a general tool to establish exponential ergodicity.

Steady-state analysis of load-balancing algorithms in the sub-Halfin–Whitt regime

We study a class of load-balancing algorithms for many-server systems (N servers). Each server has a buffer of size $b-1$ with $b=O(\sqrt{\log N})$, i.e. a server can have at most one job in service and $b-1$ jobs queued. We focus on the steady-state performance of load-balancing algorithms in the heavy traffic regime such that the load of the system is $\lambda = 1 - \gamma N^{-\alpha}$ for $0<\alpha<0.5$ and $\gamma > 0,$ which we call the sub-Halfin–Whitt regime ($\alpha=0.5$ is the so-called Halfin–Whitt regime). We establish a sufficient condition under which the probability that an incoming job is routed to an idle server is 1 asymptotically (as $N \to \infty$) at steady state. The class of load-balancing algorithms that satisfy the condition includes join-the-shortest-queue, idle-one-first, join-the-idle-queue, and power-of-d-choices with $d\geq \frac{r}{\gamma}N^\alpha\log N$ (r a positive integer). The proof of the main result is based on the framework of Stein’s method. A key contribution is to use a simple generator approximation based on state space collapse.

Subdiffusive Load Balancing in Time-Varying Queueing Systems

The degree to which delays or queue lengths equalize under load-balancing algorithms gives a good indication of their performance. Some of the most well-known results in this context are concerned with the asymptotic behavior of the delay or queue length at the diffusion scale under a critical load condition, where arrival and service rates do not vary with time. For example, under the join-the-shortest-queue policy, the queue length deviation process, defined as the difference between the greatest and smallest queue length as it varies over time, is at a smaller scale (subdiffusive) than that of queue lengths (diffusive).

Analysis of the shortest relay queue policy in a cooperative random access network with collisions

The aim of this work concerns the performance analysis of systems with interacting queues under the join the shortest queue policy. The case of two interacting queues results in a two-dimensional random walk with bounded transitions to non-neighboring states, which in turn results in complicated boundary behavior. Although this system violates the conditions of the compensation approach due to the transitions to non-neighboring states, we show how to extend the framework of the approach and how to apply it to the system at hand. Moreover, as an additional level of theoretic validation, we have compared the results obtained with the compensation approach with those obtained using the power series algorithm and we have shown that the compensation approach outperforms the power series algorithm in terms of numerical efficiency. In addition to the fundamental contribution, the results obtained are also of practical value, since our analysis constitutes a first attempt toward gaining insight into the performance of such interacting queues under the join the shortest queue policy. To fully comprehend the benefits of such a protocol, we compare its performance to the Bernoulli routing scheme as well as to that of the single relay system. Extensive numerical results show interesting insights into the system’s performance.