G.MDP@score.stanford.edu (Mike Peeler) (01/05/87)
In article <816@mtunb.UUCP> dmt@mtunb.UUCP (Dave Tutelman) writes: >In article <2066@brl-adm.ARPA> G.MDP@score.stanford.edu (Mike Peeler) writes: >>Trunk capacity increases exponentially with the number of trunk >>lines, not linearly, as at first you might expect. Figure out >>how much traffic a highway can bear as a function of its width >>in lanes. That's basically the same thing, and some fancy math >>helps (if you think statistical analysis is fancy math). > Not really true. > The highway analogy deals with a very small number of servers > in the queueing system (say, 1 to 3). In this range, I agree that > small increases in servers give large increases in capacity, IF > your standard is based on blocking probability. (I noted above > that this IS the standard used by phone companies. But it's > not the only one possible, and probably isn't the one used > by highway engineers. If, for instance, you used an average > server occupancy as your standard, capacity would vary linearly > with number of servers, for ALL server sizes.) Yes, of course I'm talking about blocking probability. Phone callers can't see such things as utilization levels, but they sure can tell if they can't get through. On highways, I would look at traffic jams, which I think behave similar to blocking probability, rather than at traffic density, which is obviously linear with the number of lanes. I exclude No-Cal (as opposed to Lo-Cal) drivers, among whom arbitrarily low densities can cause a jam. > In any event, interoffice telephone trunk groups are much larger > (tens to hundreds of servers). For this size of group, capacity > varies pretty nearly linearly with number of servers, for any > required blocking probability. Well, sure. First, from that "very small number of servers" where the exponential behavior does hold, I ought to be able to use induction to prove it for large trunk groups. Anyway, it's clear that the blocking probability is monotonically decreasing and bounded by zero, so naturally it's asymptotically linear. On the flip side, the capacity curve gets so steep that it, too, is "pretty nearly linear", for practical purposes. Buy that? Cheers, Mike -------
dmt@mtunb.UUCP (Dave Tutelman) (01/06/87)
In article <2108@brl-adm.ARPA> G.MDP@score.stanford.edu (Mike Peeler) writes: >In article <816@mtunb.UUCP> dmt@mtunb.UUCP (Dave Tutelman) writes: >>In article <2066@brl-adm.ARPA> G.MDP@score.stanford.edu (Mike Peeler) writes: >>>Trunk capacity increases exponentially with the number of trunk >>>lines, not linearly, as at first you might expect. >>>How much traffic can a highway bear as a function of its width >>>in lanes. That's basically the same thing. >> Not really true. >> The highway analogy deals with a very small number of servers >> in the queueing system (say, 1 to 3). In this range, I agree that >> small increases in servers give large increases in capacity, if >> your standard is based on blocking probability. >> [However], interoffice telephone trunk groups are much larger >> (tens to hundreds of servers). For this size of group, capacity >> varies pretty nearly linearly with number of servers, for any >> required blocking probability. > >Well, sure. First, from that "very small number of servers" >where the exponential behavior does hold, I ought to be able to >use induction to prove it for large trunk groups. Anyway, it's >clear that the blocking probability is monotonically decreasing >and bounded by zero, so naturally it's asymptotically linear. >On the flip side, the capacity curve gets so steep that it, too, >is "pretty nearly linear", for practical purposes. Buy that? Yup! My point is that for "interesting" blocking probabilities (1%-5%) and "interesting" trunk group sizes ( >10), we are operating in the linear, not the exponential, part of the curve. The exponential growth for 1% blocking flattens off at about 4 servers (big highway, vanishingly small trunk group). For 10 servers, you'd have to engineer for blocking probabilities below .0001 (I'm not sure how far below; my tables didn't show it). This is an interesting blocking level for certain problems (say, server = Air Traffic Controller), but not public telephone trunk groups. Since this discussion arose from the question, "Do modems cost the phone company more plant investment than voice?" let me note how this is relevant. I took Mike's comment that "trunk capacity increases exponentially with number of trunk lines" to mean "a tiny investment by the phone company results in a large capacity increase." (Forgive me if that's not what you implied; it is what I inferred.) In fact, given the size and traffic levels of typical trunk groups, the cost is roughly proportional to the increased load. > >Cheers, > Mike >------- Likewise, Dave