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BPS 800.000 8 700.000 600.000 500.000 400.000 300.000 200.000 100.000 0 BPS 400.000 300.000 200.000 100.000 11 12 13 14 a Hours 0 BPS 160.000 140.000 120.000 100.000 80.000 60.000 40.000 20.000 11 12 13 14 Hours b 0 11 12 13 14 Hours c Figure 10 Time variation of a data sample with different integration times a) 1 second integration time b) 10 seconds integration time c) 1 minute integration time. Backup transfers are often programmed to be performed at night time. Video transmissions are usually one-way, with long-lasting high bitrate transmission, and tend to be mostly evening entertainment. The transmission speed of speech is normalised to a fixed bit rate, whereas data traffic bit rates vary over an extremely broad range. These and other peculiarities may lead to stochastic variations as well as short- and long-time profiles that are quite different from those of telephone traffic. However, the activities of business hours and leisure hours are in many ways decisive for profiles even for other types of traffic than telephone. And the basic laws of the traffic theory are invariant, while the environments, the conditions and the relevant parameters are highly variable. If the independence assumption no longer applies, correlation must be considered, and the analysis is much more complicated. Some diagrams showing real traffic observations are presented in the following section. 7 Some traffic observations As an illustration two diagrams of real traffic as observed at the main telephone exchange in central Trondheim are shown in Figures 8 and 9. Both figures present daily profiles of 10 minute averages between hours 7 and 24, and averaged over several days in order to get smoothed samples demonstrating typical features. Figure 8 contains the results of 25,000 calls from typical residential subscribers. The main features are a slow growth from virtually zero between 7 and 8 hours, rising to a morning peak around 10 (maximum activity), a fall off until a lower peak between 15 and 16 (towards end of working hours), a marked low around 17 (supper, nap), before increasing again to an evening peak (leisure time) of the same size as the morning peak. The further gradual fall-off until midnight is broken by a sharp drop around the evening news. The corresponding profile in Figure 9 contains 48,000 calls from business and residential telephones as well. Here, business telephones dominate during working hours and residential telephones during the evening. The main difference is found between 10 and 16 hours, showing a marked drop during lunch hour before an afternoon peak between 13 and 14 of
t1 t2 t3 t4 t5 t6 T1 T2 T3 T4 T5 T6 T7 equal size as the one around 10. One observation is that the residential subscribers cause a better utilisation of the system than do the business subscribers, due to the greater traffic concentration of the latter. An example of data traffic observations [3] is shown in Figure 10, a), b), and c). The observations are given in the form of bits per second from a workstation over a three hour period. Three degrees of resolution are given, 1 second, 10 seconds and 1 minute, clearly indicating the loss of detail by integration over longer time. Indicative is the reduction of peak values from 750 via 350 to 160 kb/s for the same set of data. (Note different ordinate scales.) 8 Traffic modelling τ1 τ2 τ3 τ4 τ6 τ5 s1 s2 s3 s5 Figure 11 The double process of arrivals and departures Up till now we have focused on the traffic load on a set of servers and on different types of variation. Apart from an indication that the traffic is generated by calls from traffic sources we have not studied how traffic is created. In Figure 3 the call arrivals create all the +1 steps. In the long run there are equally many -1 steps, and these all stem from departures from the system. Earlier or later any arrival must result in a departure. This double process may be depicted as in Figure 11. If we start by observing only arrivals, we have the points T1 to T7 with the intervals t1 to t6 . Arrivals and departures are always connected in pairs. This implies that the longterm averages of arrival rate λ and departure rate γ must be equal. That again secures a limitation of the traffic load as long as λ and the holding times (service times si ) are limited. It is seen from the figure that the traffic measured during the period T = T7 – T1 is a little less than 1 Erlang. This is seen by adding the holding times and dividing the sum by T: A = (s1 s4 s6 + s2 + s3 + s4 + s5 + s6)/T. By reducing the intervals T i – T i–1 , thereby increasing λ, the traffic increases. The same effect is obtained by increasing s i . 9 Little’s formula, the “Ohm’s law” of teletraffic It is well known from electrophysics that the three quantities voltage v, current i and resistance r are connected by the formula v = i ⋅ r, and that in an electric network this formula may be applied to the whole network or any part of it. This is Ohm’s law. In a similar way the three quantities traffic A, arrival rate λ and service (holding) time s are connected by the formula A = λ⋅s (3) This is Little’s formula, and like Ohm’s law in an electric network, Little’s formula applies to the whole or any part of a traffic network. A difference is that for Ohm’s law constant values are assumed, whereas Little’s formula applies to mean values, including the constant case (given co-ordination of arrivals and departures, which is a bit artificial). The only condition is a stationary process, which will be discussed later. In simple terms stationarity means that the statistical properties of the involved processes remain unchanged over time. There are no other conditions laid on the statistical distributions of the variables. With reference to the traffic models of Figures 1 and 2 an alternative model is shown in Figure 12. The following indexes can be used: o ⇒ offered, relating to sources l ⇒ lost q ⇒ queue c ⇒ carried, relating to servers. One could also use corresponding indexing on holding times, however we use, according to common practice, the following: s ⇒ service time w ⇒ waiting time (in queue). Very often h is used for holding time. It can be indexed to show which system part it is related to. For instance hl may be used to indicate that even a lost call occupies a source for a non-zero time. According to Little’s formula we obtain: Traffic load on queue: Aq = λq ⋅ w = λc ⋅ w Traffic load on server group: Ac = λc ⋅ s Traffic load on source group (encompassing the whole system): Ao = (λo – λl ) ⋅ (w + s) + λl ⋅ hl = λc ⋅ (w + s) + λl ⋅ hl We see that, if the lost traffic holding time hl is zero, the traffic load on the sources, Ao , is actually equal to the sum of the queue traffic and the server traffic. In this case – but not in general – the “non-empty” call rate is identical for all parts of the system, whereas the holding times are different. This would be different if the calls waiting in queue had a limited patience, so that some of them would leave the queue without being served. That would reduce the mean holding time on the source group and the queue, and the arrival rate on the server group. Thus the load would be reduced on all parts of the system. The linear property of Little’s formula can be expressed by A = A1 + A2 = λ1 ⋅ s1 + λ2 ⋅ s2 , (4) N sources X X X X Traffic requests X X X X X X X X X Overflow Carried calls n servers O O O O O O O q waiting positions Figure 12 Traffic model with overflow from queue 9
Page 2 and 3: Contents Feature Guest editorial, A
Page 4 and 5: 2 costs will be such that decisions
Page 6 and 7: 4 N sources 1 The delay along the p
Page 8 and 9: 6 Sunday Monday Tuesday Wednesday T
Page 12 and 13: 10 where the contributions from dif
Page 14 and 15: 12 1.75 1.50 1.25 1.00 0.75 0.50 0.
Page 16 and 17: 14 Table 2 Survey of some distribut
Page 18 and 19: 16 0 Local calls mean: 27 sec. Std.
Page 20 and 21: 18 Frame 5 Equilibrium calculations
Page 22 and 23: 20 j k λ ij µji λ ik µ ki λ ir
Page 24 and 25: 22 14 Disturbed and shaped traffic
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Page 28 and 29: 26 n * ooo---o A * M, V stream is n
Page 30 and 31: 28 A-subscr. Network B-subscr. λ(
Page 32 and 33: 30 Frame 6 Basic queuing model From
Page 34 and 35: 32 Traffic sources (N) ⇒ III - -
Page 36 and 37: 34 Since Gw (t) is the survivor fun
Page 38 and 39: 36 - Mean response time depends onl
Page 40 and 41: 38 Figure 38 Mixed international da
Page 42 and 43: 40 3 Gaaren, S. LAN interconnection
Page 44 and 45: 42 Solution of some problems in the
Page 46 and 47: 44 Table 3 a. The “Loss” (in
Page 48 and 49: 46 Table 6. (x = 3). a n 0.0 0.1 0.
Page 50 and 51: 48 Telephone Systems” (published
Page 52 and 53: 50 The life and work of Conny Palm
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Page 56 and 57: 54 of random traffic it is tacitly
Page 58 and 59: 56 Architectures for the modelling
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58 QoS user Service catagories user
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60 (N)-service-user (N)-service-pro
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62 ACSE Presentation layer Session
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64 Traffic source 1 Traffic source
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oth the traditional ack-based and t
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68 With this as a basis, some modif
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70 300 250 200 150 100 50 0 100 90
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72 Erl kErl 50 40 30 20 10 0 8 10 1
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74 Si: extreme high n s n s normal
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76 cost increase which must be cove
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78 with two to four hours read-out,
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80 Similarly as for Poisson-traffic
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82 throughput 1 Introduction Commun
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84 processor load Figure 4 its cust
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86 load control mechanisms. It is u
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88 ers to use the same facility at
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90 CE CE Figure 3 IRIM with cluster
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92 and one mini IRSU and one normal
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Table 4 Combined signalling and pac
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96 mission cost of large capacities
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40 30 20 10 98 TE-network TE-region
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100 change the routing for the next
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102 According to our plans reroutin
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104 Based on the experiences with t
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user access device 106 sound/speech
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108 overs for mobile stations that
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110 radio interface tion inside a b
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Figure 9 One way of defining space
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114 frequency code here. However, t
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116 class 1 class 2 class 3 [1,1,1]
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1.00E-00 1.00E-01 1.00E-02 case I c
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400 300 200 100 25 25 20 10 120 (a)
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Erlang Erlang Erlang 122 7 6 5 4 3
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124 Table 6 Call holding times (in
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Registered number of calls Register
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Table 9 Number of call attempts, su
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130 LAN Interconnection Traffic Mea
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132 gering table could be employed
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134 OSI name/layer 7) Application 6
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136 kbit/s 4000 3000 2000 1000 0 kb
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138 - The external LAN traffic is e
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Number of type 2 connections 140 Fe
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142 Number of type 2 connections No
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144 Number of type 2 connections Fi
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146 Preliminary studies indicate th
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148 Background Traffic Test Traffic
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150 ATM cell header 2.1.1 Measuring
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152 δ τ Figure 8 Deterministic in
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154 and CMR measurements was set to
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156 Sparc2 (SunOS 4.1.1) or Sparc10
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158 Segment size [byte] 8192 6144 4
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160 Throughput [Mbit/s] Throughput
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162 Segment size [byte] Unacknowled
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164 Throughput [Mbit/s] 30 20 4 kby
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166 Throughput [Mbit/s] Throughput
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168 TEL A TEL B Satellitedish Swiss
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170 Cell Discard Ratio Cell Discard
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172 Number of Type 2 Sources Number
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174 Synthetic load generation for A
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176 The type of the modulated proce
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178 Transp. Network LLC MAC PHY Tra
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180 0.005 0.004 0.003 0.002 0.001 0
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182 Number of active sources of the
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184 that these times are identicall
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186 Level duration The state sojour
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188 are summarised, before potentia
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190 Figure 4.4 Excerpt from the ATM
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192 4.2.6 Load extremes By specifyi
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194 14th international teletraffic
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196 how this change in “event rat
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advantage is that it is generally v
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200 ˜X = X + β(C − E(C)) (5.1)
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202 Table 3 Case 1: N = 4, λ/µ =
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204 A(t) 1 x - A on off Ti Pji Pij
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206 6.1.3 Control variables The num
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208 Some important queuing models f
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210 � � � A(ζ) � ζn �
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212 The main reason for giving (2.5
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214 M 0 0 1 For the sake of simplic
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216 cell-loss where I is the contou
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218 cell-loss 10 0 10 -1 10 -2 10 -
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220 Notes on a theorem of L. Takác
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222 and (25) (26) The parameters ck
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224
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226 Documents types that are prepar
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228 The rules for preparation and a
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230 Terminals/ Terminal Adapters
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232 plaintext Figure 1 The PNO Ciph
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