140:, a value (which can be a non-integer such as 43.5) followed by “erlangs” represents the average number of concurrent calls carried by the circuits (or other service-providing elements), where that average is calculated over some reasonable period of time. The period over which the average is calculated is often one hour, but shorter periods (e.g., 15 minutes) may be used where it is known that there are short spurts of demand and a traffic measurement is desired that does not mask these spurts. One erlang of carried traffic refers to a single resource being in continuous use, or two channels each being in use fifty percent of the time, and so on. For example, if an office has two telephone operators who are both busy all the time, that would represent two erlangs (2 E) of traffic; or a radio channel that is occupied continuously during the period of interest (e.g. one hour) is said to have a load of 1 erlang.
147:, a value followed by “erlangs” represents the average number of concurrent calls that would have been carried if there were an unlimited number of circuits (that is, if the call-attempts that were made when all circuits were in use had not been rejected). The relationship between offered traffic and carried traffic depends on the design of the system and user behavior. Three common models are (a) callers whose call-attempts are rejected go away and never come back, (b) callers whose call-attempts are rejected try again within a fairly short space of time, and (c) the system allows users to wait in queue until a circuit becomes available.
284:
busy-hour traffic (in erlangs). This is the average number of concurrent calls during a given one-hour period of the day, where that period is selected to give the highest result. (This result is called the time-consistent busy-hour traffic). An alternative is to calculate a busy-hour traffic value separately for each day (which may correspond to slightly different times each day) and take the average of these values. This generally gives a slightly higher value than the time-consistent busy-hour value.
128:
the possibility of an unlimited queue and it gives the probability that a new call will need to wait in the queue due to all servers being in use. Erlang's formulae apply quite widely, but they may fail when congestion is especially high causing unsuccessful traffic to repeatedly retry. One way of accounting for retries when no queue is available is the
Extended Erlang B method.
2405:, where congestion breeds further congestion at peak times. In such cases, it is first necessary for many additional circuits to be made available so that the high loss can be alleviated. Once this action has been taken, congestion will return to reasonable levels and Erlang's equations can then be used to determine how exactly many circuits are really required.
2297:
2408:
An example of an instance which would cause such a High Loss System to develop would be if a TV-based advertisement were to announce a particular telephone number to call at a specific time. In this case, a large number of people would simultaneously phone the number provided. If the service provider
2400:
When Erlang developed the Erlang-B and Erlang-C traffic equations, they were developed on a set of assumptions. These assumptions are accurate under most conditions; however in the event of extremely high traffic congestion, Erlang's equations fail to accurately predict the correct number of circuits
283:
The practical measurement of traffic is typically based on continuous observations over several days or weeks, during which the instantaneous traffic is recorded at regular, short intervals (such as every few seconds). These measurements are then used to calculate a single result, most commonly the
166:
have broad applicability beyond telephony. They apply wherever users arrive more or less at random to receive exclusive service from any one of a group of service-providing elements without prior reservation, for example, where the service-providing elements are ticket-sales windows, toilets on an
127:
A distinguishing assumption behind the Erlang B formula is that there is no queue, so that if all service elements are already in use then a newly arriving call will be blocked and subsequently lost. The formula gives the probability of this occurring. In contrast, the Erlang C formula provides for
2110:
servers. However, if all the servers are busy when a request arrives from a source, the request is queued. An unlimited number of requests may be held in the queue in this way simultaneously. This formula calculates the probability of queuing offered traffic, assuming that blocked calls stay in
521:
depend on the number of active sources. The total number of sources is assumed to be infinite. The Erlang B formula calculates the blocking probability of a buffer-less loss system, where a request that is not served immediately is aborted, causing that no requests become queued. Blocking occurs
104:
Carried traffic in erlangs is the average number of concurrent calls measured over a given period (often one hour), while offered traffic is the traffic that would be carried if all call-attempts succeeded. How much offered traffic is carried in practice will depend on what happens to unanswered
501:
and is not limited to telephone networks, since it describes a probability in a queuing system (albeit a special case with a number of servers but no queueing space for incoming calls to wait for a free server). Hence, the formula is also used in certain inventory systems with lost sales.
2115:, for a specified desired probability of queuing. However, the Erlang C formula assumes that callers never hang up while in queue, which makes the formula predict that more agents should be used than are really needed to maintain a desired service level.
154:, expressed as a certain number of erlangs, meaning the exact number of calls taking place at a point in time. In this case the number is a non-negative integer. Traffic-level-recording devices, such as moving-pen recorders, plot instantaneous traffic.
167:
airplane, or motel rooms. (Erlang's models do not apply where the service-providing elements are shared between several concurrent users or different amounts of service are consumed by different users, for instance, on circuits carrying data traffic.)
124:. His results, which are still used today, relate quality of service to the number of available servers. Both formulae take offered load as one of their main inputs (in erlangs), which is often expressed as call arrival rate times average call length.
1154:
510:, so call arrival instants are independent. Further, it is assumed that the message lengths (holding times) are exponentially distributed (Markovian system), although the formula turns out to apply under general holding time distributions.
179:(QoS). For example, in a system where there is no queuing, the GoS may be that no more than 1 call in 100 is blocked (i.e., rejected) due to all circuits being in use (a GoS of 0.01), which becomes the target probability of call blocking,
677:
2121:
1446:, which is not always a good match, but is valid for any statistical distribution of call holding times with a finite mean. It applies to traffic transmission systems that do not buffer traffic. More modern examples compared to
2069:
expresses the probability that an arriving customer will need to queue (as opposed to immediately being served). Just as the Erlang B formula, Erlang C assumes an infinite population of sources, which jointly offer traffic of
935:
170:
The goal of Erlang's traffic theory is to determine exactly how many service-providing elements should be provided in order to satisfy users, without wasteful over-provisioning. To do this, a target is set for the
2669:
Inayatullah, M., Ullah, F.K., Khan., A.N., 'An
Automated Grade Of Service Measuring System', IEEE—ICET 2006, 2nd International Conference on Emerging Technologies, Peshawar, Pakistan 13–14 November 2006, pp.
505:
The formula applies under the condition that an unsuccessful call, because the line is busy, is not queued or retried, but instead really vanishes forever. It is assumed that call attempts arrive following a
421:
For a situation where the traffic to be handled is completely new traffic, the only choice is to try to model expected user behavior. For example, one could estimate active user population,
1469:
differs from the classic Erlang-B assumptions by allowing for a proportion of blocked callers to try again, causing an increase in offered traffic from the initial baseline level. It is an
294:, is measured on an already overloaded system, with a significant level of blocking, it is necessary to take account of the blocked calls in estimating the busy-hour offered traffic
1458:(OPS). Erlang B was developed as a trunk sizing tool for telephone networks with holding times in the minutes range, but being a mathematical equation it applies on any time-scale.
493:
that describes the probability of call losses for a group of identical parallel resources (telephone lines, circuits, traffic channels, or equivalent), sometimes referred to as an
1019:
1736:
1676:
1892:
1833:
1025:
790:
267:
1958:
1587:
116:. In Erlang's analysis of efficient telephone line usage he derived the formulae for two important cases, Erlang-B and Erlang-C, which became foundational results in
2374:
2057:
and the recall factor can be used to calculate the probability that all of a caller's attempts are lost, not just their first call but also any subsequent retries.
2055:
1985:
1922:
1786:
1614:
1554:
1498:
710:
522:
when a new request arrives at a time where all available servers are currently busy. The formula also assumes that blocked traffic is cleared and does not return.
2292:{\displaystyle P_{w}={{{\frac {E^{m}}{m!}}{\frac {m}{m-E}}} \over \left(\sum \limits _{i=0}^{m-1}{\frac {E^{i}}{i!}}\right)+{\frac {E^{m}}{m!}}{\frac {m}{m-E}}}\,}
1527:
562:
2345:
2323:
2108:
2088:
2028:
2005:
1759:
517:
servers (such as telephone lines). The rate expressing the frequency at which new calls arrive, λ, (birth rate, traffic intensity, etc.) is constant, and does
39:
2660:'Designing optimal voice networks for businesses, government, and telephone companies' by J. Jewett, J. Shrago, B. Yomtov, TelCo Research, Chicago, 1980.
796:
2111:
the system until they can be handled. This formula is used to determine the number of agents or customer service representatives needed to staff a
465: erlangs. (The division by 60 translates the busy-hour call/transaction arrival rate into a per-minute value, to match the units in which
2740:
2573:
101:
has the capacity to be used for 60 minutes in one hour. Full utilization of that capacity, 60 minutes of traffic, constitutes 1 erlang.
2692:
2608:
2518:
747:
This may be expressed recursively as follows, in a form that is used to simplify the calculation of tables of the Erlang B formula:
513:
The Erlang B formula assumes an infinite population of sources (such as telephone subscribers), which jointly offer traffic to
2581:, Transactions of the Danish Academy of Technical Sciences, vol. 2, Akademiet for de Tekniske Videnskaber, archived from
2624:
Guoping Zeng (June 2003), "Two common properties of the erlang-B function, erlang-C function, and Engset blocking function",
202:
34:
2409:
had not catered for this sudden peak demand, extreme traffic congestion will develop and Erlang's equations cannot be used.
201:, based on different models of user behavior and system operation. These may each be derived by means of a special case of
2780:
2484:
2651:
Messerli, E.J., 1972. 'Proof of a convexity property of the Erlang B formula'. Bell System
Technical Journal 51, 951–953.
2775:
2558:
536:
that a new call arriving to the resources group is rejected because all resources (servers, lines, circuits) are busy:
2712:"Kennedy I., School of Electrical and Information Engineering, University of the Witwatersrand, Personal Communication"
2444:
1447:
97:
or carried load on service-providing elements such as telephone circuits or telephone switching equipment. A single
2711:
2434:
2790:
2385:
736:
is a dimensionless load unit calculated as the mean arrival rate, λ, multiplied by the mean call holding time,
1455:
206:
1451:
280:
are expressed using the same units of time (seconds and calls per second, or minutes and calls per minute).
117:
20:
2741:"Solution of some Problems in the Theory of Probabilities of Significance in Automatic Telephone Exchanges"
974:
399:(counting successful calls and blocked calls), and the average call-holding time (for successful calls),
1690:
1625:
322:). For this purpose, where the system includes a means of counting blocked calls and successful calls,
301:(which is the traffic value to be used in the Erlang formulae). The offered traffic can be estimated by
1844:
433:(proportion of daily activity that will fall in the busy hour), and average holding time/service time,
2785:
2459:
1149:{\displaystyle {\frac {1}{B(E,j)}}=1+{\frac {j}{E}}{\frac {1}{B(E,j-1)}}\ \forall {j}=1,2,\ldots ,m.}
1794:
2449:
2424:
2418:
965:
498:
163:
113:
69:
2755:
2538:, Montreux, 26–31 October: International Telephone Consultative Committee, 1946, pp. 60–62,
753:
744:
to prove that the erlang unit has to be dimensionless for Little's Law to be dimensionally sane.
497:. It is, for example, used to dimension a telephone network's links. The formula was derived by
243:
176:
86:
2582:
1589:, each of which accounts for the recalls arising from the previously calculated offered traffic
2544:
2688:
2680:
2604:
2552:
2514:
1930:
1559:
2633:
2539:
672:{\displaystyle P_{b}=B(E,m)={\frac {\frac {E^{m}}{m!}}{\sum _{i=0}^{m}{\frac {E^{i}}{i!}}}}}
526:
172:
2352:
2033:
1963:
1900:
1764:
1592:
1532:
1476:
688:
2381:
1443:
1435:
507:
121:
2454:
1506:
741:
329:
can be estimated directly from the proportion of calls that are blocked. Failing that,
2439:
2330:
2308:
2093:
2073:
2013:
1990:
1744:
1556:, which is successively adjusted to calculate a sequence of new offered traffic values
198:
2638:
2601:
Probability, statistics, and queueing theory : with computer science applications
2769:
718:
is the number of identical parallel resources such as servers, telephone lines, etc.
930:{\displaystyle B(E,j)={\frac {EB(E,j-1)}{EB(E,j-1)+j}}\ \forall {j}=1,2,\ldots ,m.}
98:
94:
50:
437:(expressed in minutes). A projection of busy-hour offered traffic would then be
429:(number of calls/transactions per user per day), busy-hour concentration factor,
112:
named the international unit of telephone traffic the erlang in 1946 in honor of
2464:
2429:
2389:
2112:
494:
2488:
556:
identical parallel resources (servers, communication channels, traffic lanes).
1619:
1. Calculate the probability of a caller being blocked on their first attempt
1348:"""Calculate the probability of call losses."""
19:
This article is about the measurement unit in telephony. For other uses, see
1470:
90:
213:
method provides a further traffic solution that draws on Erlang's results.
395:
in an overloaded system is to measure the busy-hour call arrival rate,
2715:
2388:, therefore the Erlang C formula follows from the assumptions of the
1473:
rather than a formula and adds an extra parameter, the recall factor
728:
is the normalised ingress load (offered traffic stated in erlang).
109:
1503:
The steps in the process are as follows. It starts at iteration
964:) is calculated in numerical computation in order to ensure
2376:
is the probability that a customer has to wait for service.
2421:(discussing cellular network capacity in Erlang/MHz/cell)
2401:
required because of re-entrant traffic. This is termed a
2380:
It is assumed that the call arrivals can be modeled by a
2572:
Brockmeyer, E.; Halstrøm, H. L.; Jensen, Arne (1948),
2534:"Traffic handled on a circuit or group of circuits",
2355:
2333:
2311:
2124:
2096:
2076:
2036:
2016:
1993:
1966:
1933:
1903:
1847:
1797:
1767:
1747:
1693:
1628:
1595:
1562:
1535:
1509:
1479:
1442:. It requires that call arrivals can be modeled by a
1028:
977:
799:
756:
691:
565:
246:
350:
in the Erlang formula and the resulting estimate of
552:is the total offered traffic in erlang, offered to
65:
57:
46:
33:
28:
2368:
2339:
2317:
2291:
2102:
2082:
2049:
2022:
1999:
1979:
1952:
1916:
1886:
1827:
1780:
1753:
1730:
1684:2. Calculate the probable number of blocked calls
1670:
1608:
1581:
1548:
1521:
1492:
1148:
1013:
929:
784:
704:
671:
261:
2485:"How Many? A Dictionary of Units of Measurement"
2384:and that call holding times are described by an
2325:is the total traffic offered in units of erlangs
189:There are several resulting formulae, including
1529:with a known initial baseline level of traffic
221:Offered traffic (in erlangs) is related to the
287:Where the existing busy-hour carried traffic,
8:
1924:is the initial (baseline) level of traffic.
40:ITU Telecommunication Standardization Sector
162:The concepts and mathematics introduced by
132:Traffic measurements of a telephone circuit
2706:
2704:
25:
2637:
2603:. New York: Academic Press. p. 184.
2543:
2360:
2354:
2332:
2310:
2288:
2267:
2251:
2245:
2221:
2215:
2203:
2192:
2163:
2147:
2141:
2140:
2138:
2129:
2123:
2095:
2075:
2041:
2035:
2030:has been found, the blocking probability
2015:
1992:
1971:
1965:
1938:
1932:
1908:
1902:
1883:
1871:
1852:
1846:
1824:
1818:
1808:
1796:
1772:
1766:
1746:
1727:
1721:
1711:
1698:
1692:
1667:
1652:
1633:
1627:
1600:
1594:
1567:
1561:
1540:
1534:
1508:
1484:
1478:
1114:
1075:
1065:
1029:
1027:
978:
976:
895:
821:
798:
781:
755:
696:
690:
650:
644:
638:
627:
607:
600:
570:
564:
245:
1454:(OBS) and several current approaches to
1450:where Erlang B is still applicable, are
2476:
1434:The Erlang B formula is decreasing and
2632:(12–13), Elsevier Science: 1287–1296,
2550:
1987:, and iterate until a stable value of
210:
1838:4. Calculate the new offered traffic
1500:, which defines the recall attempts.
1014:{\displaystyle {\frac {1}{B(E,0)}}=1}
194:
190:
7:
1741:3. Calculate the number of recalls,
233:(the average time of a phone call),
2626:Mathematical and Computer Modelling
2189:
186:, when using the Erlang B formula.
2511:Fundamentals of Telecommunications
1927:5. Return to step 1, substituting
1761:, assuming a fixed Recall Factor,
1731:{\displaystyle B_{e}=E_{k}P_{b}\,}
1671:{\displaystyle P_{b}=B(E_{k},m)\,}
1111:
892:
485:with a hyphen), also known as the
150:A third measurement of traffic is
16:Load measure in telecommunications
14:
2685:Queueing Systems Volume 1: Theory
2575:The life and works of A.K. Erlang
2545:11.1004/020.1000/4.237.43.en.1001
2396:Limitations of the Erlang formula
1887:{\displaystyle E_{k+1}=E_{0}+R\,}
378:) to provide a first estimate of
105:calls when all servers are busy.
203:continuous-time Markov processes
1828:{\displaystyle R=B_{e}R_{f}\,}
1664:
1645:
1102:
1084:
1050:
1038:
999:
987:
877:
859:
848:
830:
815:
803:
772:
760:
712:is the probability of blocking
594:
582:
525:The formula provides the GoS (
1:
2639:10.1016/S0895-7177(03)90040-9
2536:CCIF - XIVth Plenary Assembly
2010:Once a satisfactory value of
388:Another method of estimating
2445:Erlang programming language
785:{\displaystyle B(E,0)=1.\,}
529:) which is the probability
262:{\displaystyle E=\lambda h}
217:Calculating offered traffic
2807:
2513:. John Wiley. p. 57.
2509:Freeman, Roger L. (2005).
2419:System spectral efficiency
336:can be estimated by using
18:
2754:: 5. 1917. Archived from
2435:Discrete-event simulation
425:, expected level of use,
231:average call-holding time
2557:: CS1 maint: location (
2386:exponential distribution
2347:is the number of servers
1456:optical packet switching
1309:
1158:
157:
1953:{\displaystyle E_{k+1}}
1681:as above for Erlang B.
1582:{\displaystyle E_{k+1}}
1452:optical burst switching
489:, is a formula for the
136:When used to represent
118:teletraffic engineering
2599:Allen, Arnold (1978).
2370:
2341:
2319:
2293:
2214:
2104:
2084:
2051:
2024:
2001:
1981:
1954:
1918:
1888:
1829:
1782:
1755:
1732:
1672:
1610:
1583:
1550:
1523:
1494:
1150:
1015:
940:Typically, instead of
931:
786:
706:
673:
643:
371:/(1 −
315:/(1 −
263:
143:When used to describe
2371:
2369:{\displaystyle P_{w}}
2342:
2320:
2294:
2188:
2105:
2085:
2052:
2050:{\displaystyle P_{b}}
2025:
2002:
1982:
1980:{\displaystyle E_{k}}
1955:
1919:
1917:{\displaystyle E_{0}}
1889:
1830:
1783:
1781:{\displaystyle R_{f}}
1756:
1733:
1673:
1611:
1609:{\displaystyle E_{k}}
1584:
1551:
1549:{\displaystyle E_{0}}
1524:
1495:
1493:{\displaystyle R_{f}}
1471:iterative calculation
1151:
1016:
932:
787:
707:
705:{\displaystyle P_{b}}
674:
623:
264:
152:instantaneous traffic
2781:Units of measurement
2460:Poisson distribution
2353:
2331:
2309:
2122:
2094:
2074:
2034:
2014:
1991:
1964:
1931:
1901:
1845:
1795:
1765:
1745:
1691:
1626:
1593:
1560:
1533:
1507:
1477:
1307:or a Python version
1026:
975:
797:
754:
689:
563:
491:blocking probability
403:, and then estimate
357:can then be used in
244:
2776:Network performance
2450:Erlang distribution
1522:{\displaystyle k=0}
966:numerical stability
499:Agner Krarup Erlang
487:Erlang loss formula
207:birth–death process
164:Agner Krarup Erlang
114:Agner Krarup Erlang
70:Agner Krarup Erlang
2748:Elektrotkeknikeren
2681:Kleinrock, Leonard
2366:
2337:
2315:
2289:
2100:
2080:
2047:
2020:
1997:
1977:
1950:
1914:
1884:
1825:
1778:
1751:
1728:
1668:
1606:
1579:
1546:
1519:
1490:
1146:
1011:
927:
782:
702:
669:
410:using the formula
259:
209:. The more recent
177:quality of service
87:dimensionless unit
2761:on July 19, 2011.
2340:{\displaystyle m}
2318:{\displaystyle E}
2286:
2283:
2265:
2235:
2179:
2161:
2103:{\displaystyle m}
2083:{\displaystyle E}
2023:{\displaystyle E}
2000:{\displaystyle E}
1754:{\displaystyle R}
1467:Extended Erlang B
1462:Extended Erlang B
1110:
1106:
1073:
1054:
1003:
891:
887:
667:
664:
621:
223:call arrival rate
211:Extended Erlang B
158:Erlang's analysis
75:
74:
2798:
2762:
2760:
2745:
2727:
2726:
2724:
2723:
2714:. Archived from
2708:
2699:
2698:
2677:
2671:
2667:
2661:
2658:
2652:
2649:
2643:
2642:
2641:
2621:
2615:
2614:
2596:
2590:
2589:
2588:on July 19, 2011
2587:
2580:
2569:
2563:
2562:
2556:
2548:
2547:
2531:
2525:
2524:
2506:
2500:
2499:
2497:
2496:
2487:. Archived from
2481:
2403:high-loss system
2375:
2373:
2372:
2367:
2365:
2364:
2346:
2344:
2343:
2338:
2324:
2322:
2321:
2316:
2298:
2296:
2295:
2290:
2287:
2285:
2284:
2282:
2268:
2266:
2264:
2256:
2255:
2246:
2241:
2237:
2236:
2234:
2226:
2225:
2216:
2213:
2202:
2181:
2180:
2178:
2164:
2162:
2160:
2152:
2151:
2142:
2139:
2134:
2133:
2109:
2107:
2106:
2101:
2089:
2087:
2086:
2081:
2067:Erlang C formula
2061:Erlang C formula
2056:
2054:
2053:
2048:
2046:
2045:
2029:
2027:
2026:
2021:
2006:
2004:
2003:
1998:
1986:
1984:
1983:
1978:
1976:
1975:
1959:
1957:
1956:
1951:
1949:
1948:
1923:
1921:
1920:
1915:
1913:
1912:
1893:
1891:
1890:
1885:
1876:
1875:
1863:
1862:
1834:
1832:
1831:
1826:
1823:
1822:
1813:
1812:
1787:
1785:
1784:
1779:
1777:
1776:
1760:
1758:
1757:
1752:
1737:
1735:
1734:
1729:
1726:
1725:
1716:
1715:
1703:
1702:
1677:
1675:
1674:
1669:
1657:
1656:
1638:
1637:
1615:
1613:
1612:
1607:
1605:
1604:
1588:
1586:
1585:
1580:
1578:
1577:
1555:
1553:
1552:
1547:
1545:
1544:
1528:
1526:
1525:
1520:
1499:
1497:
1496:
1491:
1489:
1488:
1430:
1427:
1424:
1421:
1418:
1415:
1412:
1409:
1406:
1403:
1400:
1397:
1394:
1391:
1388:
1385:
1382:
1379:
1376:
1373:
1370:
1367:
1364:
1361:
1358:
1355:
1352:
1349:
1346:
1343:
1340:
1337:
1334:
1331:
1328:
1325:
1322:
1319:
1316:
1313:
1303:
1300:
1297:
1294:
1291:
1288:
1285:
1282:
1279:
1276:
1273:
1270:
1267:
1264:
1261:
1258:
1255:
1252:
1249:
1246:
1243:
1240:
1237:
1234:
1231:
1228:
1225:
1222:
1219:
1216:
1213:
1210:
1207:
1204:
1201:
1198:
1195:
1192:
1189:
1186:
1183:
1180:
1177:
1174:
1171:
1168:
1165:
1162:
1155:
1153:
1152:
1147:
1118:
1108:
1107:
1105:
1076:
1074:
1066:
1055:
1053:
1030:
1020:
1018:
1017:
1012:
1004:
1002:
979:
952:) the inverse 1/
936:
934:
933:
928:
899:
889:
888:
886:
851:
822:
791:
789:
788:
783:
711:
709:
708:
703:
701:
700:
678:
676:
675:
670:
668:
666:
665:
663:
655:
654:
645:
642:
637:
620:
612:
611:
602:
601:
575:
574:
527:grade of service
479:Erlang B formula
473:Erlang B formula
461:
459:
458:
455:
452:
268:
266:
265:
260:
197:and the related
173:grade of service
93:as a measure of
89:that is used in
26:
2806:
2805:
2801:
2800:
2799:
2797:
2796:
2795:
2791:Queueing theory
2766:
2765:
2758:
2743:
2739:
2736:
2734:Further reading
2731:
2730:
2721:
2719:
2710:
2709:
2702:
2695:
2687:. p. 103.
2679:
2678:
2674:
2668:
2664:
2659:
2655:
2650:
2646:
2623:
2622:
2618:
2611:
2598:
2597:
2593:
2585:
2578:
2571:
2570:
2566:
2549:
2533:
2532:
2528:
2521:
2508:
2507:
2503:
2494:
2492:
2483:
2482:
2478:
2473:
2415:
2398:
2382:Poisson process
2356:
2351:
2350:
2329:
2328:
2307:
2306:
2272:
2257:
2247:
2227:
2217:
2187:
2183:
2182:
2168:
2153:
2143:
2125:
2120:
2119:
2092:
2091:
2072:
2071:
2063:
2037:
2032:
2031:
2012:
2011:
1989:
1988:
1967:
1962:
1961:
1934:
1929:
1928:
1904:
1899:
1898:
1867:
1848:
1843:
1842:
1814:
1804:
1793:
1792:
1768:
1763:
1762:
1743:
1742:
1717:
1707:
1694:
1689:
1688:
1648:
1629:
1624:
1623:
1596:
1591:
1590:
1563:
1558:
1557:
1536:
1531:
1530:
1505:
1504:
1480:
1475:
1474:
1464:
1444:Poisson process
1432:
1431:
1428:
1425:
1422:
1419:
1416:
1413:
1410:
1407:
1404:
1401:
1398:
1395:
1392:
1389:
1386:
1383:
1380:
1377:
1374:
1371:
1368:
1365:
1362:
1359:
1356:
1353:
1350:
1347:
1344:
1341:
1338:
1335:
1332:
1329:
1326:
1323:
1320:
1317:
1314:
1311:
1305:
1304:
1301:
1298:
1295:
1292:
1289:
1286:
1283:
1280:
1277:
1274:
1271:
1268:
1265:
1262:
1259:
1256:
1253:
1250:
1247:
1244:
1241:
1238:
1235:
1232:
1229:
1226:
1223:
1220:
1217:
1214:
1211:
1208:
1205:
1202:
1199:
1196:
1193:
1190:
1187:
1184:
1181:
1178:
1175:
1172:
1169:
1166:
1163:
1160:
1080:
1034:
1024:
1023:
983:
973:
972:
852:
823:
795:
794:
752:
751:
692:
687:
686:
656:
646:
622:
613:
603:
566:
561:
560:
534:
508:Poisson process
475:
469:is expressed.)
456:
453:
448:
447:
445:
443:
409:
394:
384:
377:
370:
363:
356:
349:
342:
335:
328:
321:
314:
307:
300:
293:
242:
241:
219:
184:
160:
145:offered traffic
138:carried traffic
134:
122:queueing theory
24:
17:
12:
11:
5:
2804:
2802:
2794:
2793:
2788:
2783:
2778:
2768:
2767:
2764:
2763:
2735:
2732:
2729:
2728:
2700:
2694:978-0471491101
2693:
2672:
2662:
2653:
2644:
2616:
2610:978-0120510504
2609:
2591:
2564:
2526:
2520:978-0471710455
2519:
2501:
2475:
2474:
2472:
2469:
2468:
2467:
2462:
2457:
2452:
2447:
2442:
2440:Engset formula
2437:
2432:
2427:
2422:
2414:
2411:
2397:
2394:
2378:
2377:
2363:
2359:
2348:
2336:
2326:
2314:
2300:
2299:
2281:
2278:
2275:
2271:
2263:
2260:
2254:
2250:
2244:
2240:
2233:
2230:
2224:
2220:
2212:
2209:
2206:
2201:
2198:
2195:
2191:
2186:
2177:
2174:
2171:
2167:
2159:
2156:
2150:
2146:
2137:
2132:
2128:
2099:
2079:
2062:
2059:
2044:
2040:
2019:
1996:
1974:
1970:
1947:
1944:
1941:
1937:
1911:
1907:
1895:
1894:
1882:
1879:
1874:
1870:
1866:
1861:
1858:
1855:
1851:
1836:
1835:
1821:
1817:
1811:
1807:
1803:
1800:
1775:
1771:
1750:
1739:
1738:
1724:
1720:
1714:
1710:
1706:
1701:
1697:
1679:
1678:
1666:
1663:
1660:
1655:
1651:
1647:
1644:
1641:
1636:
1632:
1603:
1599:
1576:
1573:
1570:
1566:
1543:
1539:
1518:
1515:
1512:
1487:
1483:
1463:
1460:
1310:
1159:
1157:
1156:
1145:
1142:
1139:
1136:
1133:
1130:
1127:
1124:
1121:
1117:
1113:
1104:
1101:
1098:
1095:
1092:
1089:
1086:
1083:
1079:
1072:
1069:
1064:
1061:
1058:
1052:
1049:
1046:
1043:
1040:
1037:
1033:
1021:
1010:
1007:
1001:
998:
995:
992:
989:
986:
982:
938:
937:
926:
923:
920:
917:
914:
911:
908:
905:
902:
898:
894:
885:
882:
879:
876:
873:
870:
867:
864:
861:
858:
855:
850:
847:
844:
841:
838:
835:
832:
829:
826:
820:
817:
814:
811:
808:
805:
802:
792:
780:
777:
774:
771:
768:
765:
762:
759:
730:
729:
719:
713:
699:
695:
680:
679:
662:
659:
653:
649:
641:
636:
633:
630:
626:
619:
616:
610:
606:
599:
596:
593:
590:
587:
584:
581:
578:
573:
569:
532:
474:
471:
441:
407:
392:
382:
375:
368:
361:
354:
347:
340:
333:
326:
319:
312:
305:
298:
291:
272:provided that
270:
269:
258:
255:
252:
249:
218:
215:
199:Engset formula
182:
159:
156:
133:
130:
73:
72:
67:
63:
62:
59:
55:
54:
53:, carried load
48:
44:
43:
37:
31:
30:
15:
13:
10:
9:
6:
4:
3:
2:
2803:
2792:
2789:
2787:
2784:
2782:
2779:
2777:
2774:
2773:
2771:
2757:
2753:
2749:
2742:
2738:
2737:
2733:
2718:on 2003-05-01
2717:
2713:
2707:
2705:
2701:
2696:
2690:
2686:
2682:
2676:
2673:
2666:
2663:
2657:
2654:
2648:
2645:
2640:
2635:
2631:
2627:
2620:
2617:
2612:
2606:
2602:
2595:
2592:
2584:
2577:
2576:
2568:
2565:
2560:
2554:
2546:
2541:
2537:
2530:
2527:
2522:
2516:
2512:
2505:
2502:
2491:on 2017-06-18
2490:
2486:
2480:
2477:
2470:
2466:
2463:
2461:
2458:
2456:
2453:
2451:
2448:
2446:
2443:
2441:
2438:
2436:
2433:
2431:
2428:
2426:
2423:
2420:
2417:
2416:
2412:
2410:
2406:
2404:
2395:
2393:
2391:
2387:
2383:
2361:
2357:
2349:
2334:
2327:
2312:
2305:
2304:
2303:
2279:
2276:
2273:
2269:
2261:
2258:
2252:
2248:
2242:
2238:
2231:
2228:
2222:
2218:
2210:
2207:
2204:
2199:
2196:
2193:
2184:
2175:
2172:
2169:
2165:
2157:
2154:
2148:
2144:
2135:
2130:
2126:
2118:
2117:
2116:
2114:
2097:
2077:
2068:
2060:
2058:
2042:
2038:
2017:
2008:
2007:is obtained.
1994:
1972:
1968:
1945:
1942:
1939:
1935:
1925:
1909:
1905:
1880:
1877:
1872:
1868:
1864:
1859:
1856:
1853:
1849:
1841:
1840:
1839:
1819:
1815:
1809:
1805:
1801:
1798:
1791:
1790:
1789:
1773:
1769:
1748:
1722:
1718:
1712:
1708:
1704:
1699:
1695:
1687:
1686:
1685:
1682:
1661:
1658:
1653:
1649:
1642:
1639:
1634:
1630:
1622:
1621:
1620:
1617:
1601:
1597:
1574:
1571:
1568:
1564:
1541:
1537:
1516:
1513:
1510:
1501:
1485:
1481:
1472:
1468:
1461:
1459:
1457:
1453:
1449:
1445:
1441:
1437:
1308:
1143:
1140:
1137:
1134:
1131:
1128:
1125:
1122:
1119:
1115:
1099:
1096:
1093:
1090:
1087:
1081:
1077:
1070:
1067:
1062:
1059:
1056:
1047:
1044:
1041:
1035:
1031:
1022:
1008:
1005:
996:
993:
990:
984:
980:
971:
970:
969:
967:
963:
959:
955:
951:
947:
943:
924:
921:
918:
915:
912:
909:
906:
903:
900:
896:
883:
880:
874:
871:
868:
865:
862:
856:
853:
845:
842:
839:
836:
833:
827:
824:
818:
812:
809:
806:
800:
793:
778:
775:
769:
766:
763:
757:
750:
749:
748:
745:
743:
739:
735:
727:
723:
720:
717:
714:
697:
693:
685:
684:
683:
660:
657:
651:
647:
639:
634:
631:
628:
624:
617:
614:
608:
604:
597:
591:
588:
585:
579:
576:
571:
567:
559:
558:
557:
555:
551:
547:
543:
539:
535:
528:
523:
520:
516:
511:
509:
503:
500:
496:
495:M/M/c/c queue
492:
488:
484:
480:
472:
470:
468:
464:
451:
440:
436:
432:
428:
424:
419:
417:
413:
406:
402:
398:
391:
386:
381:
374:
367:
360:
353:
346:
339:
332:
325:
318:
311:
304:
297:
290:
285:
281:
279:
275:
256:
253:
250:
247:
240:
239:
238:
236:
232:
228:
224:
216:
214:
212:
208:
204:
200:
196:
192:
187:
185:
178:
174:
168:
165:
155:
153:
148:
146:
141:
139:
131:
129:
125:
123:
119:
115:
111:
106:
102:
100:
96:
92:
88:
84:
80:
71:
68:
64:
60:
56:
52:
49:
45:
41:
38:
36:
32:
27:
22:
2756:the original
2751:
2747:
2720:. Retrieved
2716:the original
2684:
2675:
2665:
2656:
2647:
2629:
2625:
2619:
2600:
2594:
2583:the original
2574:
2567:
2535:
2529:
2510:
2504:
2493:. Retrieved
2489:the original
2479:
2455:Little's law
2425:A. K. Erlang
2407:
2402:
2399:
2379:
2301:
2066:
2064:
2009:
1926:
1896:
1837:
1740:
1683:
1680:
1618:
1502:
1466:
1465:
1439:
1433:
1306:
961:
957:
953:
949:
945:
941:
939:
746:
742:Little's law
737:
733:
731:
725:
721:
715:
681:
553:
549:
545:
541:
537:
530:
524:
518:
514:
512:
504:
490:
486:
482:
478:
476:
466:
462:
449:
438:
434:
430:
426:
422:
420:
415:
411:
404:
400:
396:
389:
387:
379:
372:
365:
358:
351:
344:
343:in place of
337:
330:
323:
316:
309:
302:
295:
288:
286:
282:
277:
273:
271:
234:
230:
226:
222:
220:
188:
180:
169:
161:
151:
149:
144:
142:
137:
135:
126:
107:
103:
99:cord circuit
95:offered load
82:
78:
76:
51:offered load
47:Unit of
2786:Teletraffic
2465:Traffic mix
2430:Call centre
2390:M/M/c queue
2113:call centre
2090:erlangs to
205:known as a
66:Named after
35:Unit system
2770:Categories
2722:2017-10-01
2495:2008-04-20
2471:References
732:Note: The
229:, and the
2277:−
2208:−
2190:∑
2173:−
1135:…
1112:∀
1097:−
916:…
893:∀
872:−
843:−
625:∑
254:λ
175:(GoS) or
91:telephony
42:standards
2683:(1975).
2553:citation
2413:See also
1315:erlang_b
1302:Function
1161:Function
548:) where
483:Erlang-B
195:Erlang C
191:Erlang B
81:(symbol
2670:230–237
2392:model.
2302:where:
1284:ErlangB
1221:Integer
1188:Integer
1164:ErlangB
682:where:
460:
446:
85:) is a
2691:
2607:
2517:
1897:where
1436:convex
1420:return
1209:Double
1197:Double
1176:Double
1109:
890:
740:. See
734:erlang
237:, by:
79:erlang
58:Symbol
29:erlang
21:Erlang
2759:(PDF)
2744:(PDF)
2586:(PDF)
2579:(PDF)
1429:inv_b
1405:inv_b
1393:inv_b
1369:range
1351:inv_b
1342:float
1339:->
110:CCITT
2689:ISBN
2605:ISBN
2559:link
2515:ISBN
2065:The
1960:for
1448:POTS
1296:InvB
1278:Next
1263:InvB
1251:InvB
1224:InvB
1203:InvB
481:(or
477:The
276:and
120:and
108:The
77:The
2634:doi
2540:hdl
1438:in
1423:1.0
1399:1.0
1360:for
1357:1.0
1333:int
1312:def
1299:End
1290:1.0
1257:1.0
1233:For
1230:1.0
1212:Dim
1200:Dim
519:not
450:NUC
2772::
2752:13
2750:.
2746:.
2703:^
2630:37
2628:,
2555:}}
2551:{{
1788:,
1616:.
1390:):
1366:in
1245:To
1218:As
1206:As
1194:As
1185:As
1173:As
968::
960:,
948:,
779:1.
726:λh
724:=
544:,
457:60
444:=
418:.
416:λh
414:=
385:.
364:=
308:=
225:,
193:,
2725:.
2697:.
2636::
2613:.
2561:)
2542::
2523:.
2498:.
2362:w
2358:P
2335:m
2313:E
2280:E
2274:m
2270:m
2262:!
2259:m
2253:m
2249:E
2243:+
2239:)
2232:!
2229:i
2223:i
2219:E
2211:1
2205:m
2200:0
2197:=
2194:i
2185:(
2176:E
2170:m
2166:m
2158:!
2155:m
2149:m
2145:E
2136:=
2131:w
2127:P
2098:m
2078:E
2043:b
2039:P
2018:E
1995:E
1973:k
1969:E
1946:1
1943:+
1940:k
1936:E
1910:0
1906:E
1881:R
1878:+
1873:0
1869:E
1865:=
1860:1
1857:+
1854:k
1850:E
1820:f
1816:R
1810:e
1806:B
1802:=
1799:R
1774:f
1770:R
1749:R
1723:b
1719:P
1713:k
1709:E
1705:=
1700:e
1696:B
1665:)
1662:m
1659:,
1654:k
1650:E
1646:(
1643:B
1640:=
1635:b
1631:P
1602:k
1598:E
1575:1
1572:+
1569:k
1565:E
1542:0
1538:E
1517:0
1514:=
1511:k
1486:f
1482:R
1440:m
1426:/
1417:E
1414:/
1411:j
1408:*
1402:+
1396:=
1387:1
1384:+
1381:m
1378:,
1375:1
1372:(
1363:j
1354:=
1345::
1336:)
1330::
1327:m
1324:,
1321:E
1318:(
1293:/
1287:=
1281:j
1275:E
1272:/
1269:j
1266:*
1260:+
1254:=
1248:m
1242:1
1239:=
1236:j
1227:=
1215:j
1191:)
1182:m
1179:,
1170:E
1167:(
1144:.
1141:m
1138:,
1132:,
1129:2
1126:,
1123:1
1120:=
1116:j
1103:)
1100:1
1094:j
1091:,
1088:E
1085:(
1082:B
1078:1
1071:E
1068:j
1063:+
1060:1
1057:=
1051:)
1048:j
1045:,
1042:E
1039:(
1036:B
1032:1
1009:1
1006:=
1000:)
997:0
994:,
991:E
988:(
985:B
981:1
962:m
958:E
956:(
954:B
950:m
946:E
944:(
942:B
925:.
922:m
919:,
913:,
910:2
907:,
904:1
901:=
897:j
884:j
881:+
878:)
875:1
869:j
866:,
863:E
860:(
857:B
854:E
849:)
846:1
840:j
837:,
834:E
831:(
828:B
825:E
819:=
816:)
813:j
810:,
807:E
804:(
801:B
776:=
773:)
770:0
767:,
764:E
761:(
758:B
738:h
722:E
716:m
698:b
694:P
661:!
658:i
652:i
648:E
640:m
635:0
632:=
629:i
618:!
615:m
609:m
605:E
598:=
595:)
592:m
589:,
586:E
583:(
580:B
577:=
572:b
568:P
554:m
550:E
546:m
542:E
540:(
538:B
533:b
531:P
515:N
467:h
463:h
454:/
442:o
439:E
435:h
431:C
427:U
423:N
412:E
408:o
405:E
401:h
397:λ
393:o
390:E
383:o
380:E
376:b
373:P
369:c
366:E
362:o
359:E
355:b
352:P
348:o
345:E
341:c
338:E
334:b
331:P
327:b
324:P
320:b
317:P
313:c
310:E
306:o
303:E
299:o
296:E
292:c
289:E
278:λ
274:h
257:h
251:=
248:E
235:h
227:λ
183:b
181:P
83:E
61:E
23:.
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