From: Lieven Smits Subject: Re: Queuing Theory and Internet Access Date: Thu, 18 Feb 1999 20:07:58 +0100 Newsgroups: sci.math To: Chris Jenkins Keywords: modeling modem pool usage as an M/M/K/K system Chris Jenkins wrote: > > Would like some help putting up a simple example to show how more > efficient a large modem pool for internet access is compared to a small > pool. > > Considerations > > Small Modem pool allows 46 users online at one time > Large pool allows 1,000 > Each user on for an average of 30 minutes > Each user on 2 times a day between 5 pm and 1 am > Usage peaks around 10 pm > > Can I use queuing theory to predict how many users each box would > support to return no more than 5% busies Yes, I think you can. If the usage varies slowly enough to be described as a sequence of constant-usage periods, then you have what would be described in the notation of David Kendall [1] as an M/M/K/K system, that is: - exponentially distributed independent interarrival times (density lambda per hour) - exponentially distributed independent service times (density mu = 2 per hour) - K identical parallel servers - no extra queue capacity apart from the K servers This is best described by a birth-and-death process: a (continuous time, discrete state space) Markov process whose states are numbered from 0 up to the number of modems (current state = number of active users). The probability of going one state up in a small interval of time is lambda times the interval (except at maximum capacity, when it is 0); the probability of going one state down is mu times the interval (except at state 0, when it is 0). All other transitions are negligible for small time intervals. See [2] for the computation of the relevant parameters, including the probability that any given new customer gets a busy signal. References: [1] David G. Kendall, Some problems in the theory of queueus, J. Royal Statist. Soc. Series B 13 (1951), 151-173 [2] Arnold O. Allen, Probability, Statistics and Queueing Theory 2nd ed. Academic Press 1990, ISBN 0-12-051051-0 Hope this helps Lieven Smits ============================================================================== From: James Carlson Subject: Re: Internet Port Usage Queuing Theory Question Date: 02 Mar 1999 07:42:18 -0500 Newsgroups: sci.math.num-analysis cjenkins@ziplink.net (Chris Jenkins) writes: > I'm trying to put together a simple example to demonstrate why it > makes sense to build large dial in modem pools for internet access > rather than the more typical small pop locations. I recall from > college days that the answer is in queuing theory but can't recall any > of the mechanics. You might want to try using an Erlang B model for that. You need to know the average call length, the number of calls per hour, and the percentage of blocked calls you're willing to tolerate. The result will give you the number of ports needed to handle the traffic. (Alternatively, you can work backwards to find out how many calls per hour you can handle with a set number of lines.) The missing part is how to convert that into a population of users. I'm not sure about that ... > Some facts > > Small Modem Pool > > 46 access ports (46 users can dial in at once) Typical user dials in > for 2 30 minute sessions between 6 pm and 2 am in a normal > distribution around 10 pm Assuming 1% blocking, I get 34.3 Erlangs. That's 68.6 calls per hour that can be supported. -- James Carlson, Consulting S/W Engineer IronBridge Networks / 55 Hayden Avenue 71.246W Vox: +1 781 372 8132 Lexington MA 02421-7996 / USA 42.423N Fax: +1 781 372 8090 "PPP Design and Debugging" --- http://people.ne.mediaone.net/carlson/ppp