A Hybrid Architecture - Scalable Continuous Media Streaming Systems - page 342

this waiting time, additional client requests such as requests 2, 3, and so on, arrive but the
admission controller will not send additional START requests to the service node. This process
repeats when a new request arrives at time t 2 .
Based on this model, we first derive the average waiting time experienced by a START request
at the service node. For the arrival process, we assume that user requests form a Poisson arrival
process with rate λ . The proportion of client requests falling within the admission threshold is

given by

2 T R

P S = (19.6)

and these clients will be statically-admitted.
Correspondingly, the proportion of dynamically-admitted clients is equal to (1 P S ). We

assume that the resultant arrival process at the admission controller is also Poisson, with a rate
equal to

−

λ D = (1 − P S ) λ (19.7)

Referring to Figure 19.8, we observe that the time between two adjacent START requests is
composed of two parts. The first part is the waiting time for a free dynamic multicast channel,
and the second part is the time until a new dynamically-admitted client request arrives. For
the first part, we let W C (

δ ) be the average waiting time for a free dynamic multicast channel

given . To derive the second part, we first note that the mean inter-arrival time between the

two requests (request x and y in Figure 19.8) immediately before and after a free dynamic
channel becomes available, called event E , is equal to 2

δ

/λ D ,or twice the normal mean inter-
arrival time. This counter-intuitive result is due to the fact that longer interval is more likely
to be encountered by the event E . With an inter-arrival time that is exponentially distributed
with mean 1

/λ D , the length-biased mean inter-arrival time as observed by the event E will

/λ D [1]. Next we observe that the event E is equally likely to occur within the interval
between the two requests, thus the mean time until the next arrival is simply half the length of
the interval, or 1

become 2

/λ D .
Therefore, the inter-arrival time for START requests is given by

1
λ S =

1
λ D

W C ( δ ) + (19.8)

where λ S is the arrival rate for START requests. For simplicity, we assume that the arrival

process formed from START requests is also a Poisson process.
For the service time of START request, it depends on the last user joining the batch (Fig-
ure 19.7). In particular, the service time of the last user equals to the arrival time a n minus the
time t m − 1 for the previous multicast of the requested video title. The service time, denoted by s ,
can range from 0 to ( T R − 2 δ ). We assume the service time s is uniformly distributed between

0 < s < T R − 2 δ (19.9)

Therefore, the dynamicmulticast channels formamulti-server queueing systemwith Poisson
arrival and uniformly distributed service time. As no close-form solution exists for such a
queueing model, we turn to the approximation by Allen and Cunneen [2] for G/G/m queues