Decay rate for a PH/M/2 queue with shortest queue discipline

Total coliform bacteria have been chosen as the indicator organism. Coliform die-away experiments have been carried out in unpolluted sea water samples collected at about 100 m off the coastline and under controlled environmental conditions. The samples were transformed into one litre clean glass beakers which were kept at constant temperature and were exposed to the solar radiation. The membrane filter technique was used for the coliform analysis. The temperature ranged from 20 to 40° C and the dilution ratios ranged from 1/50 to 1/200. Coliform decay rate in the light has been expressed as the summation of the coliform decay rate in the dark and the decay rate due to solar radiation. The solar radiation required for 90 percent coliform removal has been found to range from 17 cal/cm2 to 40 cal/cm2 within the temperature range of 25 to 30° C. Applying the linear regression analysis two different equations have been given for the high (I>10 cal/cm2.hour) and low solar intensity ranges in order to determine the coliform decay rate constant as a function of the solar intensity. T-90 values in the light have been found to follow log-normal distribution with a median T-90 value of 32 minutes. The corresponding T-90 values in the dark were found to be 70-80 times longer. Coliform decay rate in the dark has been correlated with the temperature.

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

Mathematics of Operations Research ◽

10.1287/moor.2019.1023 ◽

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.

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