In this paper, we provide a new and effective approach for studying super-exponential solution of a retrial supermarket model with Poisson arrivals, exponential service times and exponential retrial timesand with two different probing-server numbers. We describe the retrial supermarket model as a system of differential equations by means of density-dependent jump Markov processes, and obtain an iterative algorithm for computing the fixed point of the system of differential equations. Based on the fixed point, we analyze the expected sojourn time that a tagged arriving customer spends in this system, and use numerical examples to indicate different influence of the two probing-server numbers on system performance including the fixed point and the expected sojourn time. Furthermore, we analyze exponential convergence of the current location of the retrial supermarket model to the fixed point, and apply the Kurtz Theorem to study density-dependent jump Markov process given in the retrial supermarket model, which leads to a Lipschitz condition under which the fraction measure of the retrial supermarket model weakly converges to the system of differential equations . This paper arrives at a new understanding of how the workload probing can help in load balancing jobs in retrial supermarket models .
When decomposing the total orbit into N sub-orbits (or simply orbits) related to each of N servers and through comparing the numbers of customers in these orbits, we introduce a retrial supermarket model of N identical servers, where two probing-server choice numbers are respectively designed for dynamically allocating each primary arrival and each retrial arrival into these orbits when the chosen servers are all busy. Note that the designed purpose of the two choice numbers can effectively improve performance measures of this retrial supermarket model. This paper analyzes a simple and basic retrial supermarket model of N identical servers, that is, Poisson arrivals, exponential service and retrial times. To this end, we first provide a detailed probability computation to set up an infinite-dimensional system of differential equations (or mean-field equations) satisfied by the expected fraction vector. Then, as N goes to infinity, we apply the operator semigroup to obtaining the mean-field limit (or chaos of propagation) for the sequence of Markov processes which express the state of this retrial supermarket model . Specifically, some simple and basic conditions for the mean-field limit as well as for the Lipschitz condition are established through the first two moments of the queue length in any orbit. Finally, we show that the fixed point satisfies a system of nonlinear equations which is an interesting networking generalization of the tail equations given in the M/M/1 retrial queue, and also use the fixed point to give performance analysis of this retrial supermarket model through numerical computation .
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In this paper, we provide a new and effective approach for studying super-exponential solution of a
When decomposing the total orbit into N sub-orbits (or simply orbits) related to each of N servers and through comparing the numbers of customers in these orbits, we introduce a retrial supermarket model of N identical servers, where two probing-server choice numbers are respectively designed for dynamically allocating each primary arrival and each retrial arrival into these orbits when the chosen servers are all busy. Note that the designed purpose of the two choice numbers can effectively improve performance measures of this retrial supermarket model. This paper analyzes a simple and basic
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with
of N identical servers, that is,
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times and exponential retrial timesand with two different probing-server numbers. We describe the retrial supermarket model as a
and retrial times. To this end, we first provide a detailed probability computation to set up an infinite-dimensional
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by means of density-dependent jump Markov processes, and obtain an iterative algorithm for computing the fixed point of the system of differential equations. Based on the fixed point, we analyze the expected sojourn time that a tagged arriving customer spends in this system, and use numerical examples to indicate different influence of the two probing-server numbers on system performance including the fixed point and the expected sojourn time. Furthermore, we analyze exponential convergence of
(or mean-field equations) satisfied by
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current location of the
expected fraction vector. Then, as N goes to infinity, we apply the operator semigroup to obtaining the mean-field limit (or chaos of propagation) for the sequence of Markov processes which express the state of this
R
to the fixed point, and apply the Kurtz Theorem to study density-dependent jump Markov process given in the retrial supermarket model, which leads to a Lipschitz condition under which the fraction measure of the retrial supermarket model weakly converges to the system of differential equations . This paper arrives at a new understanding of how the workload probing can help in load balancing jobs in retrial supermarket models
. Specifically, some simple and basic conditions for the mean-field limit as well as for the Lipschitz condition are established through the first two moments of the queue length in any orbit. Finally, we show that the fixed point satisfies a system of nonlinear equations which is an interesting networking generalization of the tail equations given in the M/M/1 retrial queue, and also use the fixed point to give performance analysis of this retrial supermarket model through numerical computation