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Comments on “the buffer behavior in computer communication systems”

机译：关于“计算机通信系统中的缓冲区行为”的评论

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摘要

Heines1 recently analyzed the buffer behavior in computer communication system under: 1) Poisson arrivals, 2) periodic opportunities for service, 3) random blocking of service, and 4) averaging queue length and delay time observed at customer departure times. Heines'' analysis revealed a different result to that of Hsu''s [1]. This is because the probability distribution of the buffer content derived by Hsu corresponds to the epochs of "end of service intervals," while that in Heines'' corresponds to the buffer content just after a customer departure. Using a (recently introduced) discrete state level crossing analysis [2], Heines'' result can be derived from that of Hsu''s. The intent of this letter, however, is to point out an alternate viewpoint of this model and to relate Heines'' result to some previously published results. It was observed by Heines that if a data packet arrives when the system is idle, service to this data packet may be attempted only at the end of that service interval. This phenomenon may be interpreted as follows: every time the buffer becomes empty the output channel is closed for a length of a slot time. If no data packet arrives during this slot time, the channel is once again closed down for the following slot time. This is continued until at least one data packet arrives. Then the channel will be opened for service at the end of the slot time following the data packet arrival. This model is indeed an M/G/1 queue with geometric service times and "multiple server vacations." The results for this M/G/1 are easily obtained from that of Welch''s [3] and are available in [4] and [5]. Using a straightforward translation of the results in [4] and [5] we get the Laplace Stieltjes transform W(s) of the waiting time distribution for this model as (see [6]) $tilde{W}(s) = {(f-lambda) exp (-s) (1- exp (-s)) over (lambda - s + sf) exp (-s)-(lambda - s)}, quad Re(s) > 0$ where f - = Pr {channel available during a slot time} and λ is the data packet arrival rate. Now one may use this viewpoint of this model and the level crossing analysis discussed in [5] to extent this model to accomodate bulk arrival with multitype of customers.
机译：Heines 1 最近在以下情况下分析了计算机通信系统中的缓冲区行为：1）泊松到达，2）周期性服务机会，3）服务随机阻塞，和4）平均队列长度和延迟时间客户出发时间。海涅斯的分析显示出与许氏[1]不同的结果。这是因为由Hsu得出的缓冲区内容的概率分布对应于“服务间隔结束”的时期，而在Heines中的概率分布对应于刚离开客户之后的缓冲区内容。使用（最近引入的）离散状态能级交叉分析[2]，可以从Hsu的结果中得出Heines的结果。但是，这封信的目的是指出该模型的另一种观点，并将海涅斯的结果与一些先前发表的结果联系起来。 Heines观察到，如果当系统空闲时数据包到达，则只能在该服务间隔结束时尝试对此数据包进行服务。这种现象可以解释为：每次缓冲区变空时，输出通道都会关闭一段时间。如果在此时隙内没有数据包到达，则在下一个时隙内再次关闭该通道。持续进行直到至少一个数据分组到达为止。然后，在数据包到达后的时隙结束时，将打开该通道以进行服务。该模型的确是一个具有几何服务时间和“多个服务器休假”的M / G / 1队列。这个M / G / 1的结果很容易从Welch的文献[3]中获得，可以在[4]和[5]中找到。使用[4]和[5]中结果的直接转换，我们得到该模型的等待时间分布的拉普拉斯斯蒂尔杰斯变换W（s），如下所示（见[6]）$ tilde {W}（s）= { （f-lambda）exp（-s）（1- exp（-s））超过（lambda-s + sf）exp（-s）-（lambda-s）}，四元组Re（s）> 0 $其中f -＝ Pr {在时隙期间可用的信道}，并且λ是数据分组到达率。现在，人们可以使用这种模型的观点和[5]中讨论的平交分析来扩展这种模型，以适应多种类型客户的批量到达。

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