EE 489 Telecommunication Systems Engineering University of Alberta

EE 489 – Telecommunication Systems Engineering –University of Alberta, Dept. of Electrical and Computer Engineering Models for Calculating Blocking Probabilities • System must be in a Steady State – Also called state of “statistical equilibrium” – Arrival Rate of new calls equals Departure Rate of disconnecting calls – Why? • If calls arrive faster that they depart? • If calls depart faster than they arrive? Models to follow: • Binomial (finite sources) Poisson (infinite sources) • Erlang B (infinte sources, Blocked calls cleared (BCC) • Engset (BCC, finite sources) • Erlang C (infinite sources, infinite Queue) “Blocked Calls Wait” (BCW) (both BCH) Material prepared by W. Grover (1998 -2002) 1

EE 489 – Telecommunication Systems Engineering –University of Alberta, Dept. of Electrical and Computer Engineering Binomial Traffic Model • Assumptions: – m sources – A Erlangs of offered traffic • per source: TO = A/m • probability that a specific source is busy: P(B) = A/m • Then a Binomial Distribution gives the probability that a certain number (k) of those m sources is busy: P(k) is the probability of system state k, i. e. , k servers in use Material prepared by W. Grover (1998 -2002) 2

EE 489 – Telecommunication Systems Engineering –University of Alberta, Dept. of Electrical and Computer Engineering Binomial Model (2) • Now consider if we have N servers (N<m)? – We can have at most N busy sources at a time – What about the probability of blocking? • All N servers must be busy before we have blocking • All N busy only if k≥N, therefore Material prepared by W. Grover (1998 -2002) 3

EE 489 – Telecommunication Systems Engineering –University of Alberta, Dept. of Electrical and Computer Engineering Binomial Model (3) • But what does it mean if k>N? – Impossible to have more sources busy than servers to serve them – Therefore, doesn’t accurately represent reality • In reality, P(k>N) = 0 – In this model, we have sources busy without a server • …BCH effect – Arguments in favour, however: • Acts as model of user re-try behaviour • Well, at least it’s a “conservative” (i. e. , pessimistic) model for blocking – An argument against: • if m = 300, and N = 20, do you really expect me to calculate the term “ 300 choose 19” !? • if m is “large” there must be a better way to calculate P(B)… Material prepared by W. Grover (1998 -2002) 4

EE 489 – Telecommunication Systems Engineering –University of Alberta, Dept. of Electrical and Computer Engineering Poisson Blocking Model • The Poisson model emerges as the limiting case of Binomial as m ∞ and small A/m = Mean # of Busy Sources • What is ? – Mean number of busy sources – Well that’s easy …. m (A/m) = A Gives the probability of k servers in use Material prepared by W. Grover (1998 -2002) 5

EE 489 – Telecommunication Systems Engineering –University of Alberta, Dept. of Electrical and Computer Engineering Poisson Traffic Model (2) • So now we can calculate probability of blocking: Typical notation: “P” = Poisson “A” = Offered Traffic “N” = # Servers Example: Poisson P(B) with 10 E offered to 7 servers Material prepared by W. Grover (1998 -2002) 6

EE 489 – Telecommunication Systems Engineering –University of Alberta, Dept. of Electrical and Computer Engineering The Role of Traffic Tables • Note that P(N, A) assumes N is given and we calculate the blocking from A Erlangs of offered traffic • But in a design orientation we more naturally want to soilve the incverse problem, i. e. , what should N be to serve A at not more than a target P(B). • Or, consider a g. o. s requirement of 1% blocking in a system with N=10 trunks – How much offered traffic can the system handle? • Q. How do we solve this equation for A that produces P(N, A) = 1%? – Iteratively and very carefully, or… – Use traffic tables Material prepared by W. Grover (1998 -2002) 7

EE 489 – Telecommunication Systems Engineering –University of Alberta, Dept. of Electrical and Computer Engineering (Poisson) Traffic Tables P(B)=P(N, A) N A Material prepared by W. Grover (1998 -2002) 8

EE 489 – Telecommunication Systems Engineering –University of Alberta, Dept. of Electrical and Computer Engineering Example Use of Traffic Tables P(N, A)=0. 01 N=10 A=4. 14 E A. If system with N = 10 trunks has P(B) = 0. 01: System can handle Offered traffic (A) = 4. 14 E Material prepared by W. Grover (1998 -2002) 9

EE 489 – Telecommunication Systems Engineering –University of Alberta, Dept. of Electrical and Computer Engineering Poisson Traffic Tables Material prepared by W. Grover (1998 -2002) 10

EE 489 – Telecommunication Systems Engineering –University of Alberta, Dept. of Electrical and Computer Engineering Class Problem Using Poisson traffic tables, (a) Find how many trunks are needed to serve 50 E at “P. 01” (1% Poisson blocking) (b) Answer: (c) (b) Find how many trunks are needed to serve 5 E at “P. 01” Answer: (c) Compare the trunking efficiency of the 50 E group to the 5 E group. Answer: 50 E: 5 E: (d) What general insight can you gain from this? Material prepared by W. Grover (1998 -2002) 11

EE 489 – Telecommunication Systems Engineering –University of Alberta, Dept. of Electrical and Computer Engineering Efficiency of Large Groups • What if there are N = 100 trunks? – Will they serve A = 10 x 4. 14 E = 41. 4 E with same P(B) = 1%? – No! – Traffic tables will show that A = 78. 2 E! • Why will 10 times trunks serve almost 20 times traffic? – Called efficiency of large groups: groups For N = 10, A = 4. 14 E For N = 100, A = 78. 2 E The larger the trunk group, the greater the efficiency (at target loading) We already know this effect qualitatively as “economy of scale” Material prepared by W. Grover (1998 -2002) 12

EE 489 – Telecommunication Systems Engineering –University of Alberta, Dept. of Electrical and Computer Engineering Intro to Traf. Calc Software • What if we need to calculate P(N, A) value not in traffic table? – Traf. Calc: Traf. Calc Custom-designed software • Calculates P(B) or A, or • Creates custom traffic tables Example Problems: • How do we calculate P(32, 20)? • How do we calculate the A for which P(32, A) = 0. 01? Material prepared by W. Grover (1998 -2002) 13

EE 489 – Telecommunication Systems Engineering –University of Alberta, Dept. of Electrical and Computer Engineering Erlang B Model • More sophisticated model than Binomial or Poisson – Analytically recognizes that states k>N are not possible. • Blocked Calls Cleared (BCC) • Exact for trunk groups that have reroute to alternate route if blocked • If no alternate route, blocking must be low for accuracy, otherwise reattempt effects not reflected. – i. e. , while Poisson is a pessimistic estmate (due to BCH effect), Erlang B is either exact (overflow route case) or an optimistic estimate because of its BCC assumption. • Derived using birth-death process type of analysis – See selected pages from Leonard Kleinrock, Queueing Systems Volume 1: Theory, John Wiley & Sons, 1975 Material prepared by W. Grover (1998 -2002) 14

EE 489 – Telecommunication Systems Engineering –University of Alberta, Dept. of Electrical and Computer Engineering Erlang B Birth-Death Process • Consider infinitesimally small time t during which only one arrival or departure (or none) may occur • Let be the arrival rate from an infinite pool or sources • Let = 1/h be the departure rate per call – Note: if k calls in system, departure rate is k • Steady State Diagram: 0 1 Blockage Statistical equilibrium across every up/down transition pair …… 2 2 3 (N-1) N-1 N N Immediate Service Material prepared by W. Grover (1998 -2002) 15

EE 489 – Telecommunication Systems Engineering –University of Alberta, Dept. of Electrical and Computer Engineering The Erlang B Result In this framework, P(B) is now just probability of the system being in state k = N “the all servers busy state” = time congestion = call congestion state “B” = Erlang B “N” = # Servers “A” = Offered Traffic Material prepared by W. Grover (1998 -2002) 16

EE 489 – Telecommunication Systems Engineering –University of Alberta, Dept. of Electrical and Computer Engineering Erlang B Traffic Tables B(N, A)=0. 001 Example: In a BCC system with m= sources, we can accept a 0. 1% chance of blocking in the nominal case of 40 E offered traffic. However, in the extreme case of a 20% overload, we can accept a 0. 5% chance of blocking. B(N, A)=0. 005 Q. How many outgoing trunks do we need? A=40 E N=59 Nominal design: 59 trunks A 48 E Overload design: 64 trunks A. Requirement is: 64 trunks N=64 Material prepared by W. Grover (1998 -2002) 17

EE 489 – Telecommunication Systems Engineering –University of Alberta, Dept. of Electrical and Computer Engineering Example (2) P(N, A)=0. 01 N=32 A=20. 3 E Material prepared by W. Grover (1998 -2002) 18

EE 489 – Telecommunication Systems Engineering –University of Alberta, Dept. of Electrical and Computer Engineering Compare Poisson & Erlang B at High Blocking • We recognize that Poisson and Erlang B models are only approximations but which is better? – Compare them using a 4 -trunk group offered A=10 E Erlang B Poisson How can 4 trunks handle 10 E offered traffic and be busy only 2. 6% of the time? Material prepared by W. Grover (1998 -2002) 19

EE 489 – Telecommunication Systems Engineering –University of Alberta, Dept. of Electrical and Computer Engineering P(N, A) & B(N, A) at High Blocking • Obviously, the Poisson result is far off. – 4 servers offered enough traffic to keep 10 servers busy full time (10 E) should result in much higher utilization. • Erlang B result is more believable. – All 4 trunks are busy most of the time. • What if we extend the exercise by increasing A? – Erlang B result goes to 4 E carried traffic – Poisson result goes to 0 E carried traffic! • What is going on ? – Poisson degenerates with “runaway” BCH effect • The “take-away” messages are: – Poisson only good approximation when low blocking – Use Erlang B if high blocking – Erlang B is exact for “High usage” tunks groups with overflow Material prepared by W. Grover (1998 -2002) 20

EE 489 – Telecommunication Systems Engineering –University of Alberta, Dept. of Electrical and Computer Engineering Engset Traffic Model • BCC model with finite number of sources (m > N) = mean departure rate per call = mean arrival rate of a single source k = arrival rate if in the system is state k Blockage k = (m-k) m P 0 (m-1) P 1 0 (m-2) P 2 1 [M-(N-2)] …… 2 2 3 [m-(N-1)] PN-1 (N-1) PN N N Immediate Service Material prepared by W. Grover (1998 -2002) 21

EE 489 – Telecommunication Systems Engineering –University of Alberta, Dept. of Electrical and Computer Engineering Understanding the Engset Traffic Model • Think “Erlang B but with finite sources” • The key difference in the model derivation is that now the pool for further arrivals is depleted by each advance to a higher number of servers in use in the sytem state model. e. g. Erlang B = Engset (k) = m-(k-1) m P 0 P 1 0 (m-2) (m-1) P 2 1 Blockage m in all states [M-(N-2)] …… 2 2 3 [m-(N-1)] PN-1 (N-1) PN N N Immediate Service Material prepared by W. Grover (1998 -2002) 22

EE 489 – Telecommunication Systems Engineering –University of Alberta, Dept. of Electrical and Computer Engineering Engset Traffic Tables M = 30 sources # trunks (N) Traffic offered (A) P(B)=E(m, N, A) N=10 A=4. 8 E Example: 30 terminals each provide 0. 16 Erlangs to a concentrator with a goal of less than 1% blocking. P(B)<0. 01 How many outgoing trunks do we need? A = 30 x 0. 16 = 4. 8 E Check m < 10 x N? M=30 < 10 x 10 = 100 Requirement: N = 10 Trunks Material prepared by W. Grover (1998 -2002) 23

EE 489 – Telecommunication Systems Engineering –University of Alberta, Dept. of Electrical and Computer Engineering When to use Engset not Erlang B? • A general guideline is when the number of sources is actually know, M , and an Erlang B solution itself produces a solution N, for which M < 10 N. – i. e. , the numbers of sources is only ~10 x the number of servers, or less. • Will it matter much? – The difference between Erlang B and Engset may be only 1 or 2 trunks. – Wont matter much if the “servers” are just time-slots in a transmission system – Could mean a huge savings if the “servers” are human operators or, say, channels over a geo-synchronous satellite to a community in the far north. Material prepared by W. Grover (1998 -2002) 24

EE 489 – Telecommunication Systems Engineering –University of Alberta, Dept. of Electrical and Computer Engineering Erlang C “Delay-Working” Model • BCW model with infinite sources (m) and infinite queue length = arrival rate of new calls = mean departure rate per call P 0 P 1 0 1 P 2 2 2 3 …… P Blockage N N N PQ 1 N PQ 2 N …… N Immediate Service Material prepared by W. Grover (1998 -2002) 25

EE 489 – Telecommunication Systems Engineering –University of Alberta, Dept. of Electrical and Computer Engineering Erlang C Traffic Tables # trunks (N) N=18 P(B)=C(N, A) Traffic offered (A) A=7 E C(18, 7)=0. 0004 Example: What is the probability of blocking in an Erlang C system with 18 servers offered 7 Erlangs of traffic? Material prepared by W. Grover (1998 -2002) 26

EE 489 – Telecommunication Systems Engineering –University of Alberta, Dept. of Electrical and Computer Engineering Delay in Erlang C • Expected number of calls in the queue? Recall: Also: Material prepared by W. Grover (1998 -2002) 27

EE 489 – Telecommunication Systems Engineering –University of Alberta, Dept. of Electrical and Computer Engineering Problems • Q. What is the Carried Traffic of any Erlang C system? • Use Trafcalc to find out how much traffic can be offered to 5 servers with less than 10% probability of encountering delay. • How realistic is Erlang Cs aspect of an infinite Q from what you might already know of router technology? • A link in a packet switching network is observed to operate at 95% utilization and have an arrival rate of 10 packets / msec. What is the probability of the queuing delay > 5 msec? Material prepared by W. Grover (1998 -2002) 28

EE 489 – Telecommunication Systems Engineering –University of Alberta, Dept. of Electrical and Computer Engineering Comparison of Traffic Models Erlang C (BCW, sources) Poisson (BCH, sources) Erlang B (BCC, sources) Binomial (BCH, m sources) Engset (BCC, m sources) P(B) Offered Traffic (A) Material prepared by W. Grover (1998 -2002) 29

EE 489 – Telecommunication Systems Engineering –University of Alberta, Dept. of Electrical and Computer Engineering Efficiency of Large Groups • Already seen that for same P(B), increasing servers results in more than proportional increase in traffic carried example 1: and example 2: and example 3: and • What does this mean? – If it’s possible to collect together several diverse sources, you can • provide better gos at same cost, or • provide same gos at cheaper cost Material prepared by W. Grover (1998 -2002) 30

EE 489 – Telecommunication Systems Engineering –University of Alberta, Dept. of Electrical and Computer Engineering Efficiency of Large Groups (2) • Two trunk groups offered 5 Erlangs each, and B(N, A)=0. 002 5 E NN 11=13 =? How many trunks total? From traffic tables, find B(13, 5) 0. 002 5 E =? NN 22=13 Ntotal = 13 + 13 = 26 trunks Trunk efficiency? 38. 4% utilization Material prepared by W. Grover (1998 -2002) 31

EE 489 – Telecommunication Systems Engineering –University of Alberta, Dept. of Electrical and Computer Engineering Efficiency of Large Groups (3) • One trunk group offered 10 Erlangs, and B(N, A)=0. 002 How many trunks? 10 E N=20 N=? From traffic tables, find B(20, 10) 0. 002 N = 20 trunks Trunk efficiency? 49. 9% utilization For same gos, we can save 6 trunks! Material prepared by W. Grover (1998 -2002) 32

EE 489 – Telecommunication Systems Engineering –University of Alberta, Dept. of Electrical and Computer Engineering Efficiency of Large Groups (4) A B=0. 1 B=0. 01 B=0. 001 N N Material prepared by W. Grover (1998 -2002) 33

EE 489 – Telecommunication Systems Engineering –University of Alberta, Dept. of Electrical and Computer Engineering Sensitivity to Overload • Consider 2 cases: Case 1: N = 10 and B(N, A) = 0. 01 B(10, 4. 5) 0. 01, so can carry 4. 5 E What if 20% overload (5. 4 E)? B(10, 5. 4) 0. 03 3 times P(B) with 20% overload Case 1: N = 30 and B(N, A) = 0. 01 B(30, 20. 3) 0. 01, so can carry 20. 3 E What if 20% overload (24. 5 E)? B(30, 24. 5) 0. 08 8 times P(B) with 20% overload! “Trunk Group Splintering” • if high possibility of overloads, small groups may be better Material prepared by W. Grover (1998 -2002) 34

EE 489 – Telecommunication Systems Engineering –University of Alberta, Dept. of Electrical and Computer Engineering Incremental Traffic Carried by Nth Trunk • If a trunk group is of size N-1, how much extra traffic can it carry if you add one extra trunk? – Before, can carry: TC 1 = A x [1 -(B(N-1, A)] – After, can carry: TC 2 = A x [1 -(B(N, A)] • What does this mean? for very low blocking – Random Hunting: Hunting Increase in trunk group’s total carried traffic after adding an Nth trunk – Sequential Hunting: Hunting Actual traffic carried by the Nth trunk in the group Material prepared by W. Grover (1998 -2002) 35

EE 489 – Telecommunication Systems Engineering –University of Alberta, Dept. of Electrical and Computer Engineering Incremental Traffic Carried by Nth Trunk (2) Actual Traffic Carried by the Nth Trunk Extra Total Traffic Carried by adding Nth Trunk Random Hunting Sequential Hunting Material prepared by W. Grover (1998 -2002) 36

EE 489 – Telecommunication Systems Engineering –University of Alberta, Dept. of Electrical and Computer Engineering Incremental Traffic Carried by Nth Trunk (3) Note the context here! We add trunks to a group and assume that additional offered traffic is available to keep the blocking fixed at, say, B. 01 Fixed B(N, A) AN 0. 8 e. g. For B. 01 at ~ N=20, the incremental traffic carried by the group is almost asymptotic at 0. 8 Erlangs 20 N Material prepared by W. Grover (1998 -2002) 37

EE 489 – Telecommunication Systems Engineering –University of Alberta, Dept. of Electrical and Computer Engineering Example • Individual trunks are only economic if they can carry 0. 4 E or more. A trunk group of size N=10 is offered 6 E. Will all 10 trunks be economical? At least the 10 th trunk is not economical Material prepared by W. Grover (1998 -2002) 38

EE 489 – Telecommunication Systems Engineering –University of Alberta, Dept. of Electrical and Computer Engineering In-Class Example Material prepared by W. Grover (1998 -2002) 39