5.5 The Poisson Formula

5.5 The Poisson Formula

The number of arrivals per unit time at a service location can vary randomly according to one or many probability distributions. The Poisson distribution is a discrete probability distribution because it relates to the number of arrivals per unit time. The general model or formula for this probability distribution is given by the following equation:

where:

• r = number of arrivals

• p(r) = probability of arrivals

• n = mean of arrival rate

• e = base of natural logarithms (2.71828)

• r! = r factorial = r(r ˆ’ 1)*(r ˆ’ 2)* *3 * 2 * 1

The Poisson distribution corresponds to the assumption of random arrivals because each arrival is assumed to be independent of other arrivals and also independent of the state of the system. One interesting characteristic of the Poisson distribution is that its mean is equal to its variance. This means that by specifying the mean of the distribution, the entire distribution is specified.

5.5.1 Access Controller Sizing

As an example of the application of the Poisson distribution, let us consider an access controller where user calls arrive at a rate of two per unit period of time. From the Poisson formula, we obtain:

Substituting the values 0, 1, 2, , 9 for r, we obtain the probability of arrivals listed in Table 5.14, rounding to four decimal places. The probability of arrivals in excess of nine per unit period of time can be computed but is a very small value and was thus eliminated from consideration.

Table 5.14: Poisson Distribution Arrival (Rate of Two Per Unit Time)

Number of Arrivals per Period	Probability
	0.1358
1	0.2707
2	0.2707
3	0.1805
4	0.0902
5	0.0361
6	0.0120
7	0.0034
8	0.0009
9	0.0002

The probability of the arrival rate being less than or equal to some specific number, n, is the sum of the probabilities of the arrival rate being 0, 1, 2, , n. This can be expressed mathematically as follows :

This can be expressed in sigma notation (the mathematical shorthand for expressing "sums" of numbers ) as:

To determine the probability of four or fewer arrivals per unit period of time, we obtain:

From the preceding , almost 95 percent of the time four or fewer calls will arrive at the access controller at the same time, given an arrival rate or traffic intensity of 2. The probability that a number of calls in excess of four arrives during the period is equal to 1 minus the probability of four or fewer calls arriving, which is the grade of service. Thus, the grade of service when a traffic intensity of 2 erlangs is offered to four ports is:

If four calls arrive and are being processed , any additional calls are lost and cannot be handled by the access controller. The probability of this occurring is 0.0522 for a four-channel access controller, given a traffic intensity of 2 erlangs. In general, when E erlangs of traffic are offered to a service area containing n channels, the probability that the service area will fail to handle the traffic is given by:

Although commonly known as the Poisson traffic formula, the preceding equation is also known as the Molina equation after the American who first applied it to traffic theory. Because the number of channels or ports is always finite, it is often easier to compute the probability that the number of channels cannot support the traffic intensity in terms of their support. This is because the probability of support plus the probability of not supporting a given traffic intensity must equal unity. Thus, we can rewrite the Molina or Poisson traffic formula as:

To facilitate equipment and facility sizing using the Poisson traffic formula, another program was written in Microsoft QuickBasic. Table 5.15 contains the program listing of the file labeled POISSON.BAS, which is provided at the referenced Web address. This program generates a data file labeled POISSON.DAT, which contains a table of grades of service for traffic intensities ranging from 0.5 to 80 erlangs when lost calls are assumed to be held and thus follow the Poisson distribution. Both files are located in the directory BASIC at the Web address mentioned in this book.

 '$DYNAMIC OPEN "A:POISSON.DAT" FOR OUTPUT AS #1 DIM FACTORIAL#(60) DIM E#(180), B#(160, 160) REM E is the offered load in Erlangs REM S is the number of ports, dial in lines or trunks REM B(i,j) contains grade of service for port or channel i with traffic j GOSUB 100 'compute factorial 1 TO 60 C = 0 FOR I = 5 TO 800 STEP 5         C = C + 1         E# = I/10         E#(C) = E#                 K = 0                 FOR S = 0 TO 59 'vary port number                         K = K + 1                         SX# = S                         D# = 0                         FOR X = 0 TO S                         D1X# = X                         D# = D# + (E# ^ D1X#)/(FACTORIAL#(X) * 2.71828 ^ E#)                         NEXT X                 B#(K, C) = 1# - D#                 B#(X, C) = ABS(B#(K, C))                 NEXT S   NEXT I   FOR I = 1 TO 160 STEP 5 ' print 32 pages 5 entries per page         PRINT #1, "                     POISSON DISTRIBUTION"        PRINT #1, "       PROBABILITY ALL PORTS ARE BUSY WHEN CALL ATTEMPTED"         PRINT #1, "                WHICH IS THE GRADE OF SERVICE"         PRINT #1,         PRINT #1, "PORT #               TRAFFIC IN ERLANGS"         PRINT #1,        PRINT #1, USING " ##.## ##.##"; E#(I); E#(I + 1);   PRINT #1, USING " ##.##"; E#(I + 2);        PRINT #1, USING " ##.## ##.##"; E#(I + 3); E#(I + 4)         PRINT #1,         FOR S = 1 TO 60                 IF B#(S, I + 4) <.00001# GOTO 50                 PRINT #1, USING "##     #.##### "; S; B#(S, I);                 PRINT #1, USING "       #.#####"; B#(S, I + 1);                 PRINT #1, USING "       #.#####"; B#(S, I + 2);                 PRINT #1, USING "       #.#####"; B#(S, I + 3);                 PRINT #1, USING "       #.#####"; B#(S, I + 4)         NEXT S 50      REM space to top of page         FOR LINECOUNT = 1 TO (66 - S)         PRINT #1,         NEXT LINECOUNT NEXT I CLOSE #1 END REM subroutine to compute factorials 100 FOR I = 1 TO 60         p# = 1                 FOR j = I TO 1 STEP -1                 p# = p# * j                 NEXT j         FACTORIAL#(I) = p#         NEXT I         FACTORIAL#(0) = 1#         RETURN 

To execute the POISSON.BAT program using Microsoft's QuickBasic, you must execute that compiler using its /AH option. That option allows dynamic arrays to exceed 64 K.

If you require the generation of a smaller set of tables or wish to alter the increments of traffic in erlangs, both modifications can be easily accomplished. Altering the number of tables requires changing the size of the E# and B# arrays, the statements FOR I = 5 to 800 STEP 5, E# = I/10, and I = 1 TO 160 STEP 5. The FOR I = 5 TO 800 STEP 5 statement controls the number of different traffic loads that will be computed. Because E# is set to I divided by 10, the program listing computes traffic from 0.5 to 80 erlangs in increments of 0.5 erlangs. Thus, you can change the FOR statement or the statement E# = I/10 or both to alter the number of traffic loads and their values. Finally, you must alter the statement I = 1 TO 160 STEP 5 if you change the previously referenced FOR statement. This is because the FOR I = 1 TO STEP 160 STEP 5 statement controls the printing of output. As included in the POISSON.BAS listing, the FOR I = 1 TO 160 STEP 5 statement results in the printing of 32 pages with five entries per page.

Similar to ERLANG .DAT, the file POISSON.DAT is available for downloading in the directory BASIC at the previously indicated Web URL. This file is in ASCII format and can be directed to your printer by the command TYPE POISSON.DAT>LPT1: or it can be imported into a word processor that is capable of reading ASCII files. The POISSON.DAT file contains grade of service computations based on a traffic intensity of 0.5 to 80 erlangs in increments of 0.5 erlangs.

5.5.2 Formula Comparison and Utilization

To contrast the difference between erlang B and Poisson formulas, let us return to the LAN access controller examples previously considered . When 6 erlangs of traffic are offered and it is desired that the grade of service should be 0.01, 13 channels are required when the erlang B formula is employed. If the Poisson formula is used, an excerpt of one of the tables produced by executing the POISSON.BAS program would appear as indicated in Table 5.16. Using this table, a grade of service of 0.01 for a traffic intensity of 6 erlangs results in a required channel capacity somewhere between 9 and 10. Rounding to the next highest number results in a requirement for 10 channels. Now let us compare what happens at a higher traffic intensity. For a traffic intensity of 10 erlangs and the same 0.01 grade of service, you will note that 19 channels will be required when the erlang B formula is used. If the Poisson formula is used, you will note that a 0.01 grade of service based on 10 erlangs of traffic requires between 18 and 19 channels. Rounding to the next highest channel results in the Poisson formula providing the same value as provided through the use of the erlang B formula.

Table 5.16: Poisson B Distribution Extracts

	Traffic in erlangs
Port No.	5.50	6.00	6.50	7.00	7.50
1	0.99591	0.99752	0.99850	0.99909	0.99945
2	0.97344	0.98265	0.98872	0.99270	0.99530
3	0.91162	0.93803	0.95696	0.97036	0.07974
4	0.79830	0.84880	0.88815	0.91823	0.94085
5	0.64248	0.71494	0.77633	0.82701	0.86794
6	0.47108	0.55432	0.63096	0.69929	0.75856
7	0.31396	0.39369	0.47347	0.55029	0.62184
8	0.19051	0.25602	0.32724	0.40128	0.47536
9	0.10564	0.15276	0.20842	0.27091	0.33803
10	0.05377	0.08392	0.12261	0.16950	0.22359
11	0.02525	0.04262	0.06683	0.09852	0.13776
12	0.01098	0.02009	0.03388	0.05335	0.07924
13	0.00445	0.00882	0.01602	0.02700	0.04266
14	0.00168	0.00362	0.00710	0.01281	0.02156
15	0.00060	0.00140	0.00295	0.00571	0.01026
Probability that all ports are busy when call attempted, which is the grade of service.

In general, the Poisson formula produces a more conservative sizing at lower traffic intensities than the erlang B formula. At higher traffic intensities, the results are reversed . The selection of the appropriate formula depends on how one visualizes the calling pattern of users of the communications network.

5.5.3 Economic Constraints

In the previous dimensioning exercises, the number of ports or channels selected was based on a defined level of grade of service. Although we want to size equipment to have a high efficiency and keep network users happy, we must also consider the economics of dimensioning. One method that can be used for economic analysis is the assignment of a dollar value to each erlang- hour of traffic.

For a company such as an Internet service provider, the assignment of a dollar value to each erlang-hour of traffic may be a simple matter. Here, the average revenue per one-hour session could be computed and used as the dollar value assigned to each erlang-hour of traffic. For other organizations, the average hourly usage of employees waiting service could be employed.

As an example of the economics involved in sizing, let us assume lost calls are held, resulting in traffic following a Poisson distribution, and that 7.5 erlangs of traffic can be expected during the busy hour. Let us suppose we desire initially to offer a 0.02 grade of service. From the extract of the execution of the Poisson distribution program presented in Table 5.14, between 14 and 15 channels would be required. Rounding to the highest number, 15 channels would be selected to provide the desired 0.02 grade of service, which is equivalent to one call in 50 obtaining a busy signal.

LAN access controllers normally consist of a base unit that contains a number of channels or ports and an expansion chassis into which dual-port adapter cards are normally inserted to expand the capacity of the controller. Many times, you may desire to compare the potential revenue loss in comparison to expanding the access controller beyond a certain capacity. As an example of this, consider the data in Table 5.16, which indicates that when the traffic intensity is 5.5 erlangs, a 12-channel access controller would provide an equivalent grade of service. This means that during the busy hour, 2 erlangs of traffic would be lost and the network designer could then compare the cost of three additional ports on the access controller and additional modems and dial-in lines ” if access to the access controller is over the switched network ” to the loss of revenue by not being able to service the busy-hour traffic.