Analysis of Two-level Polling System Characteristics of Exhaustive Service and Asymmetrically Gated Service
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摘要: 区分优先级的轮询服务一直是研究人员讨论并探索的热点，本文则是采用了对称性与非对称性相结合的区分优先级的两级轮询服务模型.系统依托并行方式的处理模式，既提高了轮询系统的利用率，也降低服务器在查询转换期间所耗费的时间.并且运用马尔科夫链和概率母函数的方法建立了轮询系统的数学模型，通过对数学模型的解析精确地给出了两级非对称服务系统平均排队队长及查询周期的表达式.同时，根据系统终端循环周期的二阶特性量近似相等的方法，针对两级非对称模型给出了一种平均等待时间的近似解析式.Abstract: Prioritized polling services have been the hot topics discussed and explored by researchers. In this paper, a two level polling hybrid service model based on symmetry and asymmetry is adopted. The system relies on a parallel processing mode, which not only improves the utilization rate of the polling system but also reduces the time spent by the server during query conversion. The mathematical model of the polling system is established by using the Markov chain and probability function. By analyzing the mathematical model, the expressions of the average queue length and the query period of the two-level asymmetric service system are given. At the same time, according to the method that the second-order characteristics of system cycle are approximately equal, an approximate analytical expression of average waiting time is given for the two-level asymmetric model.
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Key words:
- Asymmetry /
- priority /
- queue length /
- waiting time
1) 本文责任编委 张俊 -
图 3 普通队列平均排队队长随到达率影响变化($ N=5 $)
Fig. 3 The average queue length of ordinary queue varies with arrival rate ($ N=5 $)
图 4 中心队列平均排队队长随到达率影响变化($ N=5 $)
Fig. 4 The average queue length of central queue varies with arrival rate ($ N=5 $)
图 5 普通队列平均排队队长随服务时间影响变化($ N=5 $)
Fig. 5 Variation of average queue length with service time in ordinary queue ($ N=5 $)
图 6 中心队列平均排队队长随服务时间影响变化($ N=5 $)
Fig. 6 The average queue queue length of central queue varies with service time ($ N=5 $)
图 7 循环查询周期随系统负载影响变化($ N=5 $)
Fig. 7 Cyclic query cycle varies with system load ($ N=5 $)
图 8 普通队列平均等待时间受负载影响变化($ N=5 $)
Fig. 8 Average waiting time of ordinary queues is affected by load changes ($ N=5 $)
图 9 中心队列平均等待时间受负载影响变化($ N=5 $)
Fig. 9 The average waiting time of the central queue is affected by load changes ($ N=5 $)
表 1 服务模型的基础参数
Table 1 Basic parameters of the service model
$ i $ $ \lambda _{i} $ $ \beta _{i} $ $ \gamma _{i} $ $ \lambda _{h} $ $ \beta _{h} $ 队列号 普通队列到达率 普通队列服务时间 普通队列转换时间 中心队列到达率 中心队列服务时间 1 0.001 4 2 0.01 1 2 0.003 4 3 0.01 1 3 0.006 3 1 0.01 1 4 0.04 2 1 0.01 1 5 0.01 1 1 0.01 1 
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表 2 两种模型理论值与实验值的对比
Table 2 Comparison between theoretical values and experimental values of two models
参数 队列号 非对称门限 两级优先级非对称模型 $ \lambda_1 =0.005 $, $\lambda_{{2}} =0.005 $
$ \lambda_{{3}} =0.01$, $ \lambda_{{4}} =0.01$
$ \lambda_{{5}} =0.01$, $ \lambda_{{h}} =0.01$
$ {{\beta}} _1 =4 $, $ {{\beta}} _2 =4$
$ {{\beta}} _3 =3 $, $ {{\beta}} _4 =2$
$ {{\beta}} _5 =1 $, $ {{\beta}} _h =1$
$ {{\gamma}} _1 =2 $, $ {{\gamma}} _2 =2$
$ {{\gamma}} _3 =1 $, $ {{\gamma}} _4 =1$, $ {{\gamma}} _5 =1 $$ i $ $ g_{i} (i) $ $ {\bar{W}} _{i} $ $ g_{i} (i) $ $ {\bar{W}} _{i} $ 理论值 实验值 理论值 实验值 理论值 实验值 理论值 实验值 1 0.0389 0.0389 3.6330 3.6329 0.0393 0.0393 4.5265 4.5194 2 0.0389 0.0380 3.6330 3.6313 0.0393 0.0402 4.5265 4.5230 3 0.0778 0.0771 3.6735 3.6752 0.0787 0.0783 4.5657 4.601 4 0.0778 0.0780 3.6330 3.6386 0.0787 0.0786 4.5214 4.5284 5 0.0778 0.0786 3.5925 3.5914 0.0787 0.0795 4.4771 4.4696
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