How Many Subscribers Should Share a Modem?

On the other hand, if a bunch of subscribers stay alive by, say, pinging the server continuously (that is, issuing ping -t myisp.com at a DOS prompt), ISPs will have problems. In this article, I'll present a statistical model based on a stochastic method to quantify the aforementioned situation and provide some guidelines to obtain an optimum ratio. I must stress that the model is not comprehensive and by no means deterministic; that is, it just shows the probability of subscribing at a certain ratio without resulting in busy signals. However, the model does provide a good framework for further and more complex studies accounting for as many parameters as possible to simulate real-life situations. Finally, I'll present the C source code (available electronically; see "Resource Center," page 5) for this study so that you can run it with your own parameters.

Currently Used Ratios

At present, most ISPs use anywhere from 5:1 to 12:1 subscriber/modem ratios, depending on location and subscriber population. It also appears that 4 Kbits/sec. per dial-up is a widely accepted ratio in the ISP industry. This means that if you have a T1 connection to an upstream provider at a rate of 1544 Kbits/sec., you need approximately 1544/4.350 phone lines. Furthermore, if you share each phone line among eight subscribers, you end up with 350×8=2800 subscribers, which is a sizable population for a single T1 line upstream. Putting it to mathematical form, if U_dr is the nominal upstream data rate, then the number of phone lines required, N_ph, is:

N_ph=U_dr -- 4

The total number of subscribers, N_s, then becomes:

N_s=N_ph×8=U_dr×8=2×U_dr

-- -- 4

The factor of 2 in this equation is not a constant and has a strong dependence on the U_dr. In other words, there is a good chance that a significant number of the 128 subscribers of an ISP with 56 Kbits/ sec. upstream connection will face busy signals, whereas a much smaller fraction of 2800 subscribers of an ISP with a T1 line upstream connection would have the same experience. The following model will show why such a fairly intuitive proposition may be true.

A Statistical Model

My statistical model is fairly simple and presumes the following:

1. All users will be connected at random times during a long block of time, T.

2. During the time, T, each user will go online several times on a random basis and spend anywhere from a minimum T^min_on to a maximum T^max_on online.

3. Each online time interval is followed by an offline period that could be anywhere from a minimum T^max_off to a maximum T^max_off.

This model has certain limitations. For example, it assumes that all users have similar habits in terms of getting on the Internet: They all come home at, say, 7 pm everyday and get on the Internet several times until they go to bed at, say, 12 pm. Figure 1 is a schematic demonstration of the model, where thick lines indicate on-line intervals. Clearly, this scenario is not exactly what happens in reality: People have different social habits even in the same locality. Some are Internet crazed, while others might not even check e-mail for several days. This can also change from day to day. If a significant fraction of subscribers are local businesses, you are dealing with two blocks of time (say, from 9 am to 5 pm and from 5 pm to 12 pm), and you should also account for dinner time. However, despite all these limitations, the study comes up with very plausible numbers that encourage further analysis along the same methodology.

Following these assumptions, the mean, m, and the standard deviation, s, for the online and offline intervals can be calculated as follows:

m=T^max+T^min

-- -- -- -- -2

s=T^max-T^min

-- -- -- -- -- --

2.58×2

The factor 2.58 comes from the following property of a normal distribution:

P(m-2.58s<X£m+2.58s)=99%

where P is the probability that a random number X is within the aforementioned limits (99 percent), which covers almost all of the data. The normal distribution covers the range -·<X<·, which is not practical to compute and has to be clipped somehow. These inequalities show that we are clipping the normal distribution to include 99 percent of all possibilities. Figure 2 demonstrates the limits schematically. For every subscriber in this study, we calculated the online and offline intervals in Figure 1 by using a random-number generator. The random numbers generated were normally distributed with m=0 and s=1 (standardized random numbers). The actual time intervals, X , were then calculated by a change of variables:

X=sZ+m

where Z is the generated standardized random number, and m and s were calculated from the minimum and maximum time intervals as described previously. All numbers falling outside the 99 percent range were rejected and the random-number generation was repeated until acceptable numbers were obtained. For every subscriber, we calculated a random offline interval followed by a random online interval until we reached the total block of time, T. Once all offline and online intervals were calculated for all subscribers, we divided T by a number of samples, say, N_samp and counted how many subscribers were online at each sampling point. Figure 3 shows the calculations obtained for 1000 and 100 subscribers. We assumed that individual users spend a minimum of 1 and a maximum of 10 minutes online every time they dial-up. We also assumed that users spend a minimum of 1 and a maximum of 60 minutes offline in between their online times. The total time for user activity was assumed to be four hours. The curves indicate the probability density function, P(x) that a certain percentage of the total users, x, are online simultaneously. Hence, the two curves are comparable and the areas under them are equal since:

^{1₀ P}(x)dx=1

It is interesting that the two curves have almost the same mean value -- approximately 15 percent. This indicates that, at any given time, the maximum probability is that 15 percent of the total number of users (approximately one out of every seven users) is online. It is therefore reasonable to share one phone line between every seven users. However, with the decrease in the total number of users, the curve becomes fatter and shorter, which indicates that there is considerable probability that as many as 25 percent of the users may be online at the same time. Hence, a small ISP (planning for 100 subscribers to start with, for example) is wise to initially order more phone lines, so that as few as four subscribers share a line and no one gets annoyed with busy signals. As the customer base grows, the ISP can slow the rate of ordering of new phone lines. The model can also predict what happens when users stay alive for longer periods of time. Figure 4 shows the same curves for 1000 and 100 subscribers, only this time the maximum time a user may be online is doubled. The mean value in both cases has increased to almost 25 percent, which indicates that almost 70 percent more phone lines are required to sustain the same level of performance. This could seriously damage the ISP's profitability and even push it out of business completely.

Conclusion

Despite many predictions of doom and gloom for mom-and-pop ISPs, they continue to flourish in local communities. For a start-up ISP to take off and for an existing one to continue its profitability, it is important to order downstream phone lines conscientiously. The study presented here provides a good guideline, albeit one based on a simple model to simulate the offline and online times each subscriber might reasonably spend daily. The model provides a good starting point for more sophisticated studies accounting for many more realistic parameters, including variations in subscriber habits, type of subscriber (business or home), and so on.

I am writing the source code for this study in a Java applet so that anyone can run it with their own parameters. In the meantime, you can e-mail me at [email protected] for a copy of the source code in C.

Acknowledgment

I am indebted to Professor S.B. Pope of Cornell University who introduced me to his elegant application of stochastic methods in scientific problems. The thought of this study was originated after a discussion with my colleague, Don Donigan at E1DEV. I also benefited from many discussions with Manny Khavari, another of my colleagues at E1DEV.

DDJ