traffic system design overview...system performance (i.e., service) objectives are specified, and...
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1
Traffic SystemDesign Overview
Traffic system design is a process that considers the entire telecommunicationsystem and the interrelationship of its components. Total system and sub-system performance (i.e., service) objectives are specified, and conflicts areresolved to achieve an optimum configuration. Therefore, traffic systemdesign ensures the cost-effective dimensioning of switching and transmissionequipment (traffic-handling resources or servers) to provide the requiredservice objectives (grade of service) economically. Telephone traffic (teletraff ic)theory—drawing on many disciplines including electronics, mathematics,statistics, probability, queuing theory, reliability, and economics—is at theheart of traffic system design.
1.1 TRAFFIC UNITS
Traffic units are a measure of traffic intensity, the average traffic density duringa one-hour period. The international unit of traffic intensity is the Erlang,*where one Erlang represents a circuit occupied for one hour.
* Named for A.K. Erlang, the father of telephone traffic theory [Brockmeyer, 1948].
1
2 Chapter 1 Traffic System Design Overview
The Erlang defines the efficiency (percent occupancy) of a traffic resource andrepresents the total time in hours to carry all calls. It is the traffic unit usedexclusively in classic traffic theory.
In the North American public switched telephone network (PSTN), thestandard traffic unit is the unit call (UC), which is expressed in seconds. TheUC is defined in centum-call-seconds (CCS) or more commonly, hundred-call-seconds. Equation 1.1 gives the relationship between Erlangs and CCS. Table1-1 is an Erlang-to-CCS conversion chart for selected traffic levels up to 200Erlangs (7200 CCS).
1 Erlang = 1 call-hour = 3600 call-seconds = 36 CCS (1.1)
1.2 TRAFFIC CALCULATIONS
Before common-equipment pools such as trunk groups, signaling registers,and operator positions can be dimensioned, their busy-hour traffic intensitiesmust be determined. Trunks are assigned to serve calls on an immediate basisand are held for the duration of the call. Signaling registers, operator positions,and similar servers normally serve calls on a delayed basis and are held onlylong enough to serve their specific functions.
1.2.1 Trunk-Group Traffic
Routing plans specify a mix of direct-route and alternate-route trunkgroups to provide least-cost routing of interswitch traffic through the network.The selected routing technique determines, to some extent, the level of trafficoffered to each trunk group. Offered trunk-group traffic is the total of alltraffic offered to the group. If the trunk group were large enough, it would carryall offered traffic but such a trunk group probably would not be economical.Instead, trunk groups are engineered to block a fraction of the offered busy-hour traffic, typically one to ten percent.
Chapter 1 Traffic System Design Overview
Table 1-1. Traffic-Unit Conversion Chart
Erlangs
0.050.100.150.200.25
0.300.350.400.450.50
0.550.600.650.700.75
0.800.850.900.951.00
1.051.101.151.201.25
1.301.351.401.451.50
1.551.601.651.701.75
1.801.851.901.952.00
CCS
1.83.65.47.29.0
10.812.614.416.218.0
19.821.623.425.227.0
29.830.632.434.236.0
37.839.641.443.245.0
46.848.650.452.254.0
55.857.659.461.263.0
64.866.668.470.272.0
Erlangs
2.052.102.152.202.25
2.302.352.402.452.50
2.552.602.652.702.75
2.802.852.902.953.00
3.053.103.153.203.25
3.303.353.403.453.50
3.553.603.653.703.75
3.803.853.903.954.00
CCS
73.875.677.479.281.0
82.884.686.488.290.0
91.893.695.497.299.0
100.8102.6104.4106.2108.0
109.8111.6113.4115.2117.0
118.8120.6122.4124.2126.0
127.8129.6131.4133.2135.0
136.8138.6140.4142.2144.0
Erlangs
4.054.104.154.204.25
4.304.354.404.454.50
4.554.604.654.704.75
4.804.854.904.955.00
5.055.105.155.205.25
5.305.355.405.455.50
5.555.605.655.705.75
5.805.855.905.956.00
CCS
145.8147.6149.4151.2153.0
154.8156.6158.4160.2162.0
163.8165.6167.4169.2171.0
172.8174.6176.4178.2180.0
181.8183.6185.4187.2189.0
190.8192.6194.4196.2198.0
199.8201.6203.4205.2207.0
208.8210.6212.4214.2216.0
Erlangs
6.056.106.156.206.25
6.306.356.406.456.50
6.556.606.656.706.75
6.806.856.906.957.00
7.057.107.157.207.25
7.307.357.407.457.50
7.557.607.657.707.75
7.807.857.907.958.00
CCS
217.8219.6221.4223.2225.0
226.8228.6230.4232.2234.0
235.8237.6239.4241.2243.0
244.8246.6248.4250.2252.0
253.9255.6257.4259.2261.0
262.8264.6266.4268.2270.0
271.8273.6275.4277.2279.0
280.8282.6284.4285.2288.0
Erlangs
8.058.108.158.208.25
8.308.358.408.458.50
8.558.608.658.708.75
8.808.858.908.959.00
9.059.109.159.209.25
9.309.359.409.459.50
9.559.609.659.709.75
9.809.859.909.95
10.00
CCS
289.8291.6293.4295.2297.0
298.8300.6302.4304.2306.0
307.8309.6311.4313.2315.0
316.8318.6320.4322.2324.0
325.8327.6329.4331.2333.0
334.8336.6338.4340.2342.0
343.8345.6347.4349.2351.0
352.8354.6356.4358.2360.0
{table continues)
3
Chapter 1 Traffic System Design Overview
Table 1-1. Traffic-Unit Conversion Chart (Continued)
Erlangs
10.110.210.310.410.5
10.610.710.810.911.0
11.111.211.311.411.5
11.611.711.811.912.0
12.112.212.312.412.5
12.612.712.812.913.0
13.113.213.313.413.5
13.613.713.813.914.0
CCS
363.6367.2370.8374.4378.0
381.6385.2388.8392.4396.0
399.6403.2406.8410.4414.0
417.6421.2424.8428.4432.0
431.6439.2442.8446.4450.0
453.6457.2460.8464.4468.0
471.6475.2478.8482.4486.0
489.6493.2496.8500.4504.0
Erlangs
14.114.214.314.414.5
14.614.714.814.915.0
15.115.215.315.415.5
15.615.715.815.916.0
16.116.216.316.416.5
16.616.716.816.917.0
17.117.217.317.417.5
17.617.717.817.918.0
CCS
507.6511.2514.8518.4522.0
525.6529.2532.8536.4540.0
543.6547.2550.8554.4558.0
561.6565.2568.8572.4576.0
579.6583.2586.8590.4594.0
597.6601.2604.8608.4612.0
615.6619.2622.8626.4630.0
633.6637.2640.8644.4648.0
Erlangs
18.118.218.318.418.5
18.618.718.818.919.0
19.119.219.319.419.5
19.619.719.819.920.0
20.120.220.320.420.5
20.620.720.820.921.0
21.121.221.321.421.5
21.621.721.821.922.0
CCS
651.6654.2658.8662.4666.0
669.6673.2676.8680.4684.0
687.6691.2694.8698.4702.0
705.6709.2712.8716.2720.0
723.6727.2730.8734.4738.0
741.6745.2748.8752.4756.0
759.6763.2766.8770.4774.0
777.6781.2784.8788.4792.0
Erlangs
22.122.222.322.422.5
22.622.722.822.923.0
23.123.223.323.423.5
23.623.723.823.924.0
24.124.224.324.424.5
24.624.724.824.925.0
25.125.225.325.425.5
25.625.725.825.926.0
CCS
795.6799.2802.8806.4810.0
813.6817.2820.8824.4828.0
831.6835.2838.8842.4846.0
849.6853.2856.8860.2864.0
867.6871.2874.8878.4882.0
885.6889.2892.8896.4900.0
903.6907.2910.8914.4918.0
921.6925.2928.8932.4936.0
Erlangs
26.126.226.326.426.5
26.626.726.826.927.0
27.127.227.327.427.5
27.627.727.827.928.0
28.128.228.328.428.5
28.628.728.828.929.0
29.129.229.329.429.5
29.629.729.829.930.0
CCS
939.6943.2946.8950.4954.0
957.6961.2964.8968.4972.0
975.6979.2982.8986.4990.0
993.6997.2
1000.81004.21008.0
1011.61015.21018.81022.41026.0
1029.61033.21036.81040.41044.0
1047.61051.21054.81058.41062.0
1065.61069.21072.81076.21080.0
(table continues)
4
Chapter 1 Traffic System Design Overview
Table 1-1. Traffic-Unit Conversion Chart (Continued)
Erlangs
30.130.230.330.430.5
30.630.730.830.931.0
31.131.231.331.431.5
31.631.731.831.932.0
32.132.232.332.432.5
32.632.732.832.933.0
33.133.233.333.433.5
33.633.733.833.934.0
CCS
1083.61087.21090.81094.21098.0
1101.61105.21108.81112.41116.0
1119.61123.21126.81130.41134.0
1137.61141.21144.81148.41152.0
1155.61159.21162.81166.41170.0
1173.61177.21180.81184.41188.0
1191.61195.21198.81202.41206.0
1209.61213.21216.81220.41224.0
Erlangs
34.134.234.334.434.5
34.634.734.834.935.0
35.135.235.335.435.5
35.635.735.835.936.0
36.136.236.336.436.5
36.636.736.836.937.0
37.137.237.337.437.5
37.637.737.837.938.0
CCS
1227.61231.21234.81238.41242.0
1245.61249.21252.81256.41260.0
1263.61267.21270.81274.41278.0
1281.61285.21288.81292.41296.0
1299.61303.21306.81310.41314.0
1317.61321.21324.81328.41332.0
1335.61339.21342.81346.41350.0
1353.61357.21360.81364.41368.0
Erlangs
38.138.238.338.438.5
38.638.738.838.939.0
39.139.239.339.439.5
39.639.739.839.940.0
40.140.240.340.440.5
40.640.740.840.941.0
41.141.241.341.441.5
41.641.741.841.942.0
CCS
1371.61375.21378.81382.41386.0
1389.61393.21396.81400.41404.0
1407.61411.21414.81418.41422.0
1425.61429.21432.81436.41440.0
1443.61447.21450.81454.41458.0
1461.61465.21468.81472.41476.0
1479.61483.21486.81490.41494.0
1497.61501.21504.81508.41512.0
Erlangs
42.142.242.342.442.5
42.642.742.842.943.0
43.143.243.343.443.5
43.643.743.843.944.0
44.144.244.344.444.5
44.644.744.844.945.0
45.145.245.345.445.5
45.645.745.845.946.0
CCS
1515.61519.21522.81526.41530.0
1533.61537.21540.81544.41548.0
1551.61555.21558.81562.41566.0
1569.61573.21576.81580.41584.0
1587.61591.21594.81598.41602.0
1605.61609.21612.81616.41620.0
1623.61627.21630.81634.41638.0
1641.61645.21648.81652.41656.0
Erlangs
46.146.246.346.746.5
46.646.746.846.947.0
47.147.247.347.447.5
47.647.747.847.948.0
48.148.248.348.448.5
48.648.748.848.949.0
49.149.249.349.449.5
49.649.749.849.950.0
CCS
1659.61663.21666.81670.41674.0
1677.61681.21684.81688.41692.0
1695.61699.21702.81706.41710.0
1713.61717.41720.81724.41728.0
1731.61735.21738.81742.41746.0
1749.61753.21756.81760.41764.0
1767.61771.21774.81778.41782.0
1785.61789.21792.81796.41800.0
(table continues)
5
Chapter 1 Traffic System Design Overview
Table 1-1. Traffic-Unit Conversion Chart (Continued)
Erlangs
50.551.051.552.052.5
53.053.554.054.555.0
55.556.056.557.057.5
58.058.559.059.560.0
60.561.061.562.062.5
63.063.564.064.565.0
65.566.066.567.067.5
68.068.569.069.570.0
CCS
18181836185418721890
19081926194419621980
19982016203420522070
20882106212421422160
21782196221422322250
22682286230423222340
23582376239424122430
24482466248425022520
Erlangs
70.571.071.572.072.5
73.073.574.074.575.0
75.576.076.577.077.5
78.078.579.079.580.0
80.581.081.582.082.5
83.083.584.084.585.0
85.586.086.587.087.5
88.088.589.089.590.0
CCS
25382556257425922610
26282646266426822700
27182736275427722790
28082826284428622880
28982916293429522970
29883006302430423060
30783096311431323150
31683186320432223240
Erlangs
90.591.091.592.092.5
93.093.594.094.595.0
95.596.096.597.097.5
98.098.599.099.5100.0
101.0102.0103.0104.0105.0
106.0107.0108.0109.0110.0
111.0112.0113.0114.0115.0
116.0117.0118.0119.0120.0
CCS
32583276329433123330
33483366338434023420
34383456347434923510
35283546356435823600
36363672370837443780
38163852388839243960
39964032406841044140
41764212424842844320
Erlangs
121.0122.0123.0124.0125.0
126.0127.0128.0129.0130.0
131.0132.0133.0134.0135.0
136.0137.0138.0139.0140.0
141.0142.0143.0144.0145.0
146.0147.0148.0149.0150.0
151.0152.0153.0154.0155.0
156.0157.0158.0159.0160.0
CCS
43564392442844644500
45364572460846444680
47164752478848244860
48964932496850045040
50765112514851845220
52565292532853645400
54365472550855445580
56165652568857245760
Erlangs
161.0162.0163.0164.0165.0
166.0167.0168.0169.0170.0
171.0172.0173.0174.0175.0
176.0177.0178.0179.0180.0
181.0182.0183.0184.0185.0
186.0187.0188.0189.0190.0
191.0192.0193.0194.0195.0
196.0197.0198.0199.0200.0
CCS
57965832586859045940
59766012604860846120
61566192622862646300
63366372640864446480
65166552658866246660
66966732676868046840
68766912694869847020
70567092712871647200
6
Chapter 1 Traffic System Design Overview 7
Figure 1-1 can be used to facilitate an understanding of traffic routingterms. Interswitch traffic is routed over the primary route trunk group providedthere are idle trunks available in the group. In an alternate-routing system,blocked trunk-group traffic overflows to other alternate-route trunk groups orto final-route trunk groups as indicated by the curved arrows. Trunk groupsprovided with alternate routes are often referred to as high-usage trunk groups.Final-route trunk groups do not have alternate routes; therefore, blocked trafficin a final-route trunk group is lost.
Trunk-group traffic is the product of the number and duration of callshandled by the group. Equation 1.2 can be used to calculate trunk-group traffic,expressed in Erlangs.
A=N*Tr (1.2)
where A = Offered traffic in ErlangsN = Number of calls during the busy hour7 = Mean call-holding time in hours
Number of calls refers to the total number of calls offered to the trunkgroup. Call-holding time is the total elapsed time between seizure of a trunkto serve the call and its subsequent release. The mean call-holding time is thearithmetic average of all call-holding times, expressed in hours.
CALLINGSUBSCRIBER
Figure 1-1. Interswitch Trunk Traffic Routing Diagram
CALLEDSUBSCRIBER
PRIMARYROUTE
CENTRALOFFICE
CENTRALOFFICE
THIRD& FINALROUTE
TANDEMSWITCH
THIRD& FINALROUTE TANDEM
SWITCH
FINALROUTE
8 Chapter 1 Traffic System Design Overview
Example 1-1Determine the traffic in Erlangs and CCS for a trunk group carrying 1000calls during the busy hour with an average call-holding time of 3 minutes.
A = (1000 calls/hour)(3 min/call)(l hour/60 min) = 50 Erlangs(50 Erl)(36 CCS/Erl) = 1800 CCS
1.2,2 Server-Pool Traffic
Server pools are groups of traffic resources, such as signaling registersand operator positions, that are used on a shared basis. Service requests thatcannot be met immediately are placed in a queue and served on a first-in, first-out (FIFO) basis. Server-pool traffic is directly related to offered traffic,server-holding time, and call-attempt factor, and inversely related to call-holding time as expressed in Equation 1.3.
AT.TS-C ( U )*s 7V
where As= Server-pool traffic in ErlangsAT = Total traffic served in ErlangsTs = Mean server-holding time in hoursTc = Mean call-holding time in hoursC = Call-attempt factor (dimensionless)
Total traffic served refers to the total offered traffic that requires theservices of the specific server pool for some portion of the call. For example,a dual-tone multifrequency (DTMF) receiver pool is dimensioned to serveonly the DTMF tone-dialing portion of total switch traffic generated by DTMFsignaling sources. Table 1-2 presents representative server-holding times fortypical signaling registers as a function of the number of digits received or sent.
Table 1-2. Typical Signaling Register Holding Times in Seconds
Signaling Register
Local Dial-Pulse (DP) ReceiverLocal DTMF ReceiverIncoming MF ReceiverOutgoing MF Sender
1
3.72.3
1.01.5
Number of
4
8.35.21.4
1.9
Digits Received or Sent
7
12.88.1
1.82.3
10
17.611.02.2
2.8
11
19.112.02.33.0
Chapter 1 Traffic System Design Overview 9
The mean server-holding time is the arithmetic average of all server-holding times for the specific server pool. Equation 1.4 is a general equationto calculate mean server-holding time for calls with different holding-timecharacteristics.
Ts = wT{ + faT2 + ••• + hTn (1.4)
where Ts = Mean server-holding time in hoursTv Tv*", Tn = Individual server-holding times in hoursa, b, ••• ,k = Fractions of total traffic served (a + 6 + ••• + Jfc = 1)
Example 1-2
Determine the mean DTMF receiver-holding time for a central office (CO) wheresubscribers dial local calls using a 7-digit number and toll calls using an 11 -digitnumber. Assume that 70 percent of the calls are local calls, the remainder are tollcalls, and that the typical signaling register holding times of Table 1-2 areapplicable.
T5 = (0.7)(8.1 sec) + (0.3X12.0 sec) = 9.27 sec
Call-attempt factors are dimensionless numbers that adjust offered trafficintensity to compensate for call attempts that do not result in completed calls.Therefore, call-attempt factors are inversely proportional to the fraction ofcompleted calls as defined in Equation 1.5.
C = y (1.5)
where C = Call-attempt factor (dimensionless)k = Fraction of calls completed (decimal fraction)
Example 1-3
Table 1-3 presents representative subscriber call-attempt dispositions based onempirical data amassed in the North American PSTN. Determine the call-attemptfactor for these data, where 70.7 percent of the calls were completed (k = 0.707).
C = ~ T " " O T O T " L 4 1 41 1
10 Chapter 1 Traffic System Design Overview
Table 1-3. Typical Call-Attempt Dispositions
Call-Attempt Disposition
Call was completed
Called subscriber did not answer
Called subscriber line was busy
Call abandoned without system response
Equipment blockage or failure
Customer dialing error
Called directory number changed or disconnected
Percentage
70.7
12.7
10.1
2.6
1.9
1.60.4
Example 1-4
Using Equation 1.3, determine the server-pool traffic in CCS and Erlangs for theDTMF receivers of Example 1-2, assuming total offered busy-hour subscribertraffic of 2000 CCS, a call-attempt factor of 1.5, and a mean call-holding time of3 minutes (180 seconds).
As = (2000 CCS) ( i .5)-2 |Zj!g> = l54.5CCS
(154.5 CCS) ( ^ g P S ) =4.29 Erlangs
1.3 TRAFFIC ASSUMPTIONS
Traffic formulas are based on a set of assumptions regarding the behavior oftraffic and its sources. These assumptions are not always precisely true. Ifvariations from these assumptions are small or known to have little effect,however, they can be used with confidence.
1.3.1 General Assumptions
The following assumptions are applicable to traffic formulas in general:
• The system is in statistical equilibrium.• Connection and disconnection of sources to servers occur
instantaneously.• The anticipated traffic density is the same for all sources.• Busy sources initiate no calls.• Every source has equal access to every server (full availability).
Chapter 1 Traffic System Design Overview 11
• The number of busy servers in a group is equal to the number ofbusy sources in its group of sources.
1.3.2 Number of Sources
The number of sources that can originate calls affects the service thesesources can expect to obtain. As the number of sources increases, the effect ofadding more sources diminishes. Eventually, a point is reached where there isnegligible difference in the probability of congestion regardless of how manynew sources are added. It is this point that distinguishes between finite andinfinite sources. Traffic formulas for applications where the number of sourcesin relation to the number of servers is very large assume infinite sources (worstcase for blocking). This simplifies the mathematics and minimizes the numberof required tables.
1.33 Disposition of Blocked Calls
Many assumptions for the disposition of blocked calls (which are alsoreferred to as lost calls) have been proposed, of which the three common casesare:
• If an idle server is not immediately available, the call is cleared fromthe system and the source becomes idle. This is commonly called theblocked calls cleared assumption.
• If an idle server is not immediately available, the call is held for aninterval equal to its holding time, and then the source becomes idle. Ifan idle server becomes available during the waiting period, it will beseized and held for an interval equal to the remaining portion of itsmean holding time. This is commonly called the blocked calls heldassumption.
• If an idle server is not immediately available, the call is queued untilan idle server is available. When an idle server becomes available, itwill be seized to serve the next call in queue and held for the full call-holding time. This is commonly called the blocked calls delayedassumption.
1.3.4 Holding-Time Distributions
A negative-exponential curve usually provides a reasonable fit for thevariation in holding times encountered with nondelayed traffic-handlingresources. Substituting a constant holding time equal to the average of varyingholding times has a negligible effect for these applications. The effects of
12 Chapter 1 Traffic System Design Overview
holding-time variations may be significant, however, when predicting theduration of delays. For example, the Crommelin-Pollaczek formulas are oftenused to determine service delays for resources with essentially constantholding times, such as dial-tone markers and intertoll trunks. Molnar's DelayProbability Charts for Telephone Traffic Where the Holding Times AreConstant graphically present data for these and similar applications.
1.4 GRADE OF SERVICE
Grade of service (GOS) is defined as the probability that offered traffic will beblocked or delayed. An absolutely nonblocking system has a GOS of zero,whereas a GOS of one indicates an absolutely blocking system. That is, thecloser the grade of service is to zero, the better the system.
Every traffic problem involves three interrelated parameters: offeredtraffic, traffic-handling resources (servers), and service objective (grade ofservice). This interrelationship can be pictured as a triangle, as shown in Fig.1-2. For a given service objective (base of triangle held constant), increasingoffered traffic requires a commensurate increase in the number of servers.Similarly, decreasing the number of servers requires a corresponding decreasein the level of offered traffic.
It is important to understand that a server's GOS is a prediction of theprobability of congestion (i.e., a call is blocked or delayed) at a given level of
SERVICE OBJECTIVE
Figure 1-2. Grade of Service Concept Diagram
Chapter 1 Traffic System Design Overview 13
offered traffic, not an absolute value. That is, a trunk-group grade of serviceof 0.01 does not mean that exactly one call in a hundred will be blocked duringthe busy hour. Rather, it means that, given a large volume of traffic, theprobability of congestion will tend toward one in a hundred.
Table 1-4 lists typical grade of service specifications for traffic systemdesign. Matching loss as used in this table refers to congestion (blocking) ina switching matrix such that input and output terminations cannot be intercon-nected via the interstage links. Switching matrix matching loss is not coveredin this handbook but the author's Voice Teletraffic Systems Engineeringcontains an entire chapter on the subject.
Table 1 -4. Typical Grade of Service Specifications
Parameter Specification
Trunk group loss probability 0.010
Intraoffice line-to-line loss probability 0.020
Line-to-trunk outgoing matching loss probability 0.010
Trunk-to-line incoming matching loss probability 0.020
Trunk-to-trunk tandem matching loss probability 0.005
Probability dial tone delay exceeds 3 seconds 0.015
Probability operator answer delay exceeds 10 seconds 0.050
The traffic formulas found in this handbook, used to predict grades ofservice, are all based on probability distributions. Probability distributions arebounded by the values zero and one; therefore, a grade of service (probabilityof congestion) cannot be negative nor can it exceed unity. Because of thisproperty, the probability of a call not experiencing congestion is one minus theprobability of congestion, and vice versa. These relationships are expressed inEquations 1.6 and 1.7.
P = l - G (1-6)
Q = l - / > (1.7)
where P = Probability of congestionQ = Probability of no congestion
14 Chapter 1 Traffic System Design Overview
1.5 TRAFFIC FORMULAS AND TABLES
Table 1-5 is a selection guide for the traffic formulas contained in thishandbook as a function of their typical applications. The Poisson, Erlang B,and Erlang C formulas, based on the assumption of infinite sources, are referredto as the major traffic formulas. The Binomial and Engset formulas, based onthe assumption of finite sources, are used in lieu of the major traffic formulaswhen the number of sources is small. Figure 1 -3 is a decision tree to facilitatetraffic formula selection on the basis of the standard traffic assumptions.
Table 1-5. Traffic Formula Selection Guide
TypicalApplications
Final trunkgroups inNorth AmericanPSTN
Trunk groupsand othernondelayedserver pools
Delayedserver pools
Small PBX orremote switchtrunk groups
Small lineconcentrators
Number ofSources
Infinite
Infinite
Infinite
Finite
Finite
Blocked-CallDisposition
Held
Cleared
Delayed
Held
Cleared
Holding-TimeDistribution
Constant orexponential
Constant orexponential
Exponential
Constant orexponential
Constant orexponential
TrafficFormula
Poisson
Erlang B
Erlang C
Binomial
Engset
Representative full-availability traffic tables, selected on the basis ofcommon telephone industry practice, are provided for the Poisson, Erlang B,Erlang C, Binomial, and Engset distributions. Full availability refers to theassumption that every source has equal access to every server. This assumptionis normally true for modern traffic systems. Some older systems, however,many of which are still in use, may be limited-availability systems. Limited-availability tables, such as those found in Siemens' Telephone Traffic TheoryTables and Charts and ITT Standard Electrik's Teletraffic Engineering Manual,can be used for those systems.
15
16 Chapter 1 Traffic System Design Overview
Tabulated traffic data values in this handbook are rounded off to the least-significant digit as applicable to the specific table. For example, loss probabil-ity values have been rounded off to five decimal places. This level of accuracyshould be more than adequate for practical applications—very low lossprobabilities may indicate overdesign, which is not economically sound.
Where the parameters of a specific application do not coincide with tablevalues, interpolation can be used. However, linear interpolation techniques arenot generally satisfactory for these highly nonlinear formulas. Adequateresults may be obtained with a graphic technique using semilogarithmic(semilog) graph paper, where loss probability is plotted logarithmically alongthe ordinate (vertical axis), and offered traffic is plotted linearly along theabscissa (horizontal axis). Figure 1-4 (page 17), a comparison of typical lossprobabilities for the Poisson and Erlang B distributions, is an example of thegraphic technique. Among other things it shows that, for a given lossprobability, less traffic can be offered to a trunk group dimensioned using thePoisson distribution than to one containing the same number of trunks butdimensioned using the Erlang B distribution. That is, the Poisson distributionresults in a more conservative design.
1.6 COMPUTER PROGRAMS
Computer programs, useful for interpolating between table values or todetermine more precise values for specific applications, are provided insubsequent chapters for the Poisson, Erlang B, Erlang C, Binomial, and Engsetformulas. These programs are written in BASIC because it is an easy-to-learnlanguage and is highly standardized. It is the universal programming languagefor the personal computers found in homes as well as engineering offices. Theprograms are formatted in an interactive (i.e., dialogue) style to facilitate theuser's entry of traffic parameters and include separate lines of code for each stepin an attempt to make them more easily understood by those with little or noprogramming experience.
Readers adept at computer programming may prefer to rewrite thesetraffic programs, combining a number of steps into a single line of code.Alternatively, the programs can be converted to a language such as FOR-TRAN, which was specifically designed for computational problem solving.In any case, newly entered programs should be validated by running themagainst benchmarks, such as the examples in this book, before relying on theiroutput data.
Chapter 1 Traffic System Design Overview 17
OFFERED LOAD-ERLANGS
Figure 1-4. Graphic Comparison of Poisson and Erlang B Distributions
A word of caution—computers are subject to overflow when dealing withvery large numbers. This limitation is a function of the computer hardware andsoftware, which can only process numbers within a finite range. Overflowoften occurs when calculating traffic formulas, which typically involve calcu-lation of factorials, numbers raised to the nth power, or infinite sums. Thetraffic programs provided herein have been written to avoid overflow condi-tions where possible. Overflow may still occur, however, when calculating theloss probability for a high traffic volume offered to a large number of servers,or some combination of these or other traffic parameters is encountered.
0.1000
0.0500
0.0200
0.0100 "
0.0050
0.0020
0.0010
0.0005
0.0002
0.0001
_J
POISSON ERLANG B