Comments on “the buffer behavior in computer communication systems”

Heines
¹
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.