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

2020 ◽  
Vol 45 (3) ◽  
pp. 1069-1103
Author(s):  
Anton Braverman
Keyword(s):  
Steady State ◽  
Diffusion Limit ◽  
Fluid Limit ◽  
Time Intervals ◽  
Dimensional Diffusion ◽  
Queue Model ◽  
Join The Shortest Queue ◽  
Shortest Queue ◽  
Process Level ◽  
General Tool

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.

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

2020 ◽  
Vol 45 (4) ◽  
pp. 1535-1571 ◽  
Author(s):  
Debankur Mukherjee ◽  
Sem C. Borst ◽  
Johan S. H. van Leeuwaarden ◽  
Philip A. Whiting
Keyword(s):  
Large Scale ◽  
Asymptotic Optimality ◽  
Diffusion Limit ◽  
Fluid Limit ◽  
Large Scale Systems ◽  
Minimum Number ◽  
Join The Shortest Queue ◽  
The Difference ◽  
Shortest Queue ◽  
And Diffusion

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 load-balancing algorithms in the sub-Halfin–Whitt regime

2020 ◽  
Vol 57 (2) ◽  
pp. 578-596 ◽  
Author(s):  
Xin Liu ◽  
Lei Ying
Keyword(s):  
Steady State ◽  
Positive Integer ◽  
Load Balancing ◽  
Heavy Traffic ◽  
Sufficient Condition ◽  
Steady State Analysis ◽  
Server Systems ◽  
Join The Shortest Queue ◽  
Shortest Queue ◽  
Steady State Performance

AbstractWe 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.

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.

scholarly journals Stability and moment bounds under utility-maximising service allocations: Finite and infinite networks

2020 ◽  
Vol 52 (2) ◽  
pp. 463-490
Author(s):  
Seva Shneer ◽  
Alexander Stolyar
Keyword(s):  
Steady State ◽  
Wide Class ◽  
Continuous Time ◽  
Service Rate ◽  
Fluid Limit ◽  
Natural Conditions ◽  
Infinite Systems ◽  
Moment Bounds ◽  
Infinite Networks ◽  
State Moment

AbstractWe study networks of interacting queues governed by utility-maximising service-rate allocations in both discrete and continuous time. For finite networks we establish stability and some steady-state moment bounds under natural conditions and rather weak assumptions on utility functions. These results are obtained using direct applications of Lyapunov–Foster-type criteria, and apply to a wide class of systems, including those for which fluid-limit-based approaches are not applicable. We then establish stability and some steady-state moment bounds for two classes of infinite networks, with single-hop and multi-hop message routes. These results are proved by considering the infinite systems as limits of their truncated finite versions. The uniform moment bounds for the finite networks play a key role in these limit transitions.

Super-Fine Structure in the Critical Flow-Rate of Critical Flow Venturi Nozzles

2002 ◽  
Author(s):  
Masahiro Ishibashi
Keyword(s):  
Fine Structure ◽  
Steady State ◽  
Discharge Coefficient ◽  
Constant Pressure ◽  
Pressure Ratio ◽  
Critical Flow ◽  
Time Intervals ◽  
Critical Flow Rate ◽  
A Value ◽  
Long Time

It is shown that critical flow Venturi nozzles need time intervals, i.e., more than five hours, to achieve steady state conditions. During these intervals, the discharge coefficient varies gradually to reach a value inherent to the pressure ratio applied. When a nozzle is suddenly put in the critical condition, its discharge coefficient is trapped at a certain value then afterwards approaches gradually to the inherent value. Primary calibrations are considered to have measured the trapped discharge coefficient, whereas nozzles in applications, where a constant pressure ratio is applied for a long time, have a discharge coefficient inherent to the pressure ratio; inherent and trapped coefficients can differ by 0.03–0.04%.

Steady-State stress distributions in circumferentially notched bars subjected to creep

1982 ◽  
Vol 17 (3) ◽  
pp. 123-132 ◽  
Author(s):  
K D Al-Faddagh ◽  
R T Fenner ◽  
G A Webster
Keyword(s):  
Steady State ◽  
Stress Distribution ◽  
Stress Index ◽  
Stress Distributions ◽  
Computer Solution ◽  
Time Intervals ◽  
Throat Region ◽  
Intermediate Time ◽  
State Stress ◽  
Good Agreement

The paper describes a procedure, based on a finite element method, for calculating directly the steady-state stress distribution in circumferentially notched bars subjected to creep without the need for obtaining solutions at intermediate time intervals. Good agreement is obtained with relevant approximate plasticity solutions and with numerical calculations which approach the steady-state over a period of time from the initial elastic stress distribution. Also, the procedure is equally applicable to primary, secondary, and tertiary creep, provided the variables of stress and time are separable in the creep law. Results obtained for a range of notch geometries and values of the stress index, n, are reported. It is found for each profile that a region of approximately constant effective stress, σ, independent of n, is obtained which can be used to characterise the overall behaviour of the notch throat region when a steady-state is reached sufficiently early in life. An approximate method for estimating the maximum equivalent steady-state stress across the notch throat is also presented which does not require a computer solution.

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