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# Multiserver retrial queue with setup time and its application to data centers

• * Corresponding author

The reviewing process of this paper was handled by Yutaka Takahashi and Wuyi Yue

• This paper considers a multiserver retrial queue with setup time which is motivated from application in data centers with the ON-OFF policy, where an idle server is immediately turned off. The ON-OFF policy is designed to save energy consumption of idle servers because an idle server still consumes about 60% of its peak consumption processing jobs. Upon arrival, a job is allocated to one of available off-servers and that server is started up. Otherwise, if all the servers are not available upon arrival, the job is blocked and retries in a random time. A server needs some setup time during which the server cannot process a job but consumes energy. We formulate this model using a three-dimensional continuous-time Markov chain obtaining the stability condition via Foster-Lyapunov criteria. Interestingly, the stability condition is different from that of the corresponding non-retrial queue. Furthermore, exploiting the special structure of the Markov chain together with a heuristic technique, we develop an efficient algorithm for computing the stationary distribution. Numerical results reveal that under the ON-OFF policy, allowing retrials is more power-saving than buffering jobs. Furthermore, we obtain a new insight that if the setup time is relatively long, setting an appropriate retrial time could reduce both power consumption and the mean response time of jobs.

Mathematics Subject Classification: 60K25, 68M20, 90B22.

 Citation: • • Figure 1.  The power consumption versus retrial rate for $c = 50$

Figure 2.  The power consumption versus setup rate for $c = 30, 50$

Figure 3.  The ratio $\mathrm{E}[P]/c$ versus retrial rate for $\alpha = 1/100$

Figure 4.  Mean response time versus retrial rate for $c = 30, 50$

Figure 5.  Mean response time versus retrial rate for $c = 30, 50$

Figure 6.  The power consumption versus traffic intensity for $\mu = 1$ and $\alpha = 1/10$

Figure 7.  The power consumption versus retrial rate for $c = 50$ and $\alpha = 1/10$

Table 1.  Truncation point $N$ and $\widehat{\mathit{\boldsymbol{\pi}}}^{(N)} \mathit{\boldsymbol{e}}$ for $c = 30$ and $\mu = 1/10$

 $c=30$ $\alpha = 1/100$ $\alpha = 1/10$ $\alpha = 1$ $N$ 1203 150 59 $\widehat{\mathit{\boldsymbol{\pi}}}^{(N)} \mathit{\boldsymbol{e}}$ $1.0060\times 10^{-11}$ $1.6243\times 10^{-10}$ $2.2090\times 10^{-15}$

Table 2.  Truncation point $N$ and $\widehat{\mathit{\boldsymbol{\pi}}}^{(N)} \mathit{\boldsymbol{e}}$ for $c = 50$ and $\mu = 1/10$

 $c=50$ $\alpha = 1/100$ $\alpha = 1/10$ $\alpha = 1$ $N$ 1203 149 58 $\widehat{\mathit{\boldsymbol{\pi}}}^{(N)} \mathit{\boldsymbol{e}}$ $4.6265\times 10^{-11}$ $2.4851\times 10^{-10}$ $1.1490\times 10^{-16}$

Table 3.  Truncation point $N$ and $\widehat{\mathit{\boldsymbol{\pi}}}^{(N)} \mathit{\boldsymbol{e}}$

 $c=50$ $c=100$ $\rho$ $N$ $\widehat{\mathit{\boldsymbol{\pi}}}^{(N)} \mathit{\boldsymbol{e}}$ $N$ $\widehat{\mathit{\boldsymbol{\pi}}}^{(N)} \mathit{\boldsymbol{e}}$ 0.1 39 $2.3871 \times 10^{-19}$ 39 $1.3689 \times 10^{-25}$ 0.2 66 $3.3924 \times 10^{-15}$ 66 $1.0369 \times 10^{-17}$ 0.3 92 $3.7031 \times 10^{-13}$ 92 $3.1474 \times 10^{-14}$ 0.4 118 $9.5226 \times 10^{-12}$ 118 $4.7215 \times 10^{-12}$ 0.5 144 $1.4487 \times 10^{-10}$ 144 $2.1293 \times 10^{-10}$ 0.6 170 $1.7715 \times 10^{-09}$ 170 $5.4344 \times 10^{-09}$ 0.7 197 $1.7430 \times 10^{-08}$ 196 $1.0141 \times 10^{-07}$ 0.8 228 $1.0765 \times 10^{-07}$ 227 $1.0942 \times 10^{-06}$ 0.9 349 $1.5416 \times 10^{-06}$ 321 $7.4458 \times 10^{-07}$
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