Model of the Nodes in the Packet Network Chapter 10

1. Model of the Nodes in the Packet Network Chapter 10

2. Queuing system

3. Kendall’s notation (1) • Classification of queuing system depends on: • Structure: number of servers • Arrival stream: interarrival time distribution • Service stream: service time distribution • Queue: queue capacity, queuing strategy • Kendall’s notation: A / B / N / K / S • A: interarrival time distribution • B: service time distribution • N: number of servers • K: queue capacity (number of waiting positions) • S: number of traffic sources (population size)

4. Kendall’s notation (2) • A / B / N / K / S • Interarrival (service) time distribution (example of standard notation) • M: Markovian, i.e. exponential distribution of interarrival (service) time; • D: Deterministic, i.e. constant time intervals; • G: General, i.e. arbitrary distribution of interarrival (service) time. May include correlation; • GI: General Independent, i.e. arbitrary distribution of interarrival (service) time without correlation; • Ph: Phase type distribution of time intervals.

5. Kendall’s notation (3) • Service strategy (example of standard strategies) • FCFS: First Come – First Served, i.e. ordered queue, waiting calls are serviced in successive order; • LCFS : Last Come – First Served (also denoted as LIFO: Last In – First Out), i.e. reverse ordered queue, waiting calls are serviced in reverse successive order; • SIRO: Service In Random Order (also denoted as RS: Random Selection), i.e. all waiting calls in the queue have the same probability of being chosen for service;

6. Little’sTheorem • Basic system parameters: • L the average number of calls in the system, • W mean holding time in the system per call, • Q the average number of calls in the queue, • T mean holding time in the queue per call. L = l W Q = l T the average number of calls in the queue the average call intensity mean holding time in the queue per call = X

7. Little’sTheorem • A(t) number of arrivals at the moment t, • B(t) number of calls outgoing from thesystem at the moment t, • Z(t) =A(t) - B(t) number of calls serviced in the system at the moment t, • ti holding time of call i, serviced in the system. • Arrival and departure process in the queuing system

8. Little’sTheorem t 1 1 1 å å ò = = = l = l L Z ( t ) dt 1 t ( t ) W i i t t tl i i 0 • Average number of calls serviced in the system within the period (0,τ): • Mean number of arrivals within the period (0,τ): • Mean holding time of a call in the system:

9. One server delay system with infinite queue M/M/1/∞ QUEUE = ∞ SERVER input stream (λ) output stream μ • One server available for any call if it is not busy, • Poisson arrival process with average intensity l, • exponential service time with mean value 1/μ , • Calls are waiting according to basic service strategy FIFO (first in first out). • The queue is infinite. It means that carried traffic is equal to offered traffic and calls are not blocked.

10. M/M/1/ ∞ system • Statetransition diagram • state „0” -  the server is free, • state „1” - one call is served,no call is waiting in the queue, • state „2” -  one call is served and one call is waiting in the queue, • . . ., • state „n” -  one call is servedand n-1 calls are waiting in the queue. • · . . .,

11. M/M/1/ ∞system - Analysis solution • State transition diagram of M/M/1/∞ delay system • Local balance equation for the M/M/1/∞ system

12. M/M/1/ ∞ system - characteristics • Average number of calls in the system: • Mean holding time in the system per call (Little’s Theorem):

13. M/M/1/ ∞ system - characteristics • Average number of calls in the queue: • Mean holding time in the queueper call (Little’s Theorem):

14. M/M/1/ ∞ system - characteristics • Averagenumber of calls in the system – formula derivation:

15. System with finite queue: M/M/1/N-1 system • Statetransition diagram for M/M/1/N-1 system QUEUE = N-1 SERVER=1 input stream (λ) output stream μ

16. M/M/1/N-1 system analysis solution • Local balance equation for the system M/M/1/N

17. System M/M/1/N-1 results

18. M/M/N/∞ system • N servers available for any call if are not busy, • Poisson arrival process with average intensity l, • exponential service time with mean value 1/μ , • callsare waiting according to basic service strategy FIFO (first in first out). • The queue is infinite. It means that carried traffic is equal to offered trafficand calls are not blocked.

19. M/M/N/∞ system SERVERS= N ∞ QUEUE= output stream input stream N

20. M/M/N/∞ system N N N N N N N solution • State transition diagram of M/M/N/ delay system • Local balance equation for the M/M/N/ ∞system

21. M/M/N/∞ system: Erlang C-formula N N N N N N N • State transition diagram of M/M/N/ ∞ system • ErlangC-formula (probability that an arbitrary arriving call has to wait in the queue)

22. M/M/N/∞ system - characteristics • average number of calls in the queue: • average number of calls in the system: • where: Lbusy is average number of calls served in the system.

23. M/M/N/∞ system - characteristics • mean holding time in the queue per call (Little’s Theorem): • mean holding time in the system per call (Little’s Theorem):

24. M/M/N/∞ system - characteristics • M/M/N/∞system connection with M/M/N/0 system (Erlang formula for full availability group) where a=A/N

25. M/M/N/m system • N servers available for any call if its are not busy, • Poisson arrival process with average intensity l, • exponential service time with mean value 1/μ , • callsare waiting according to basic service strategy FIFO (first in first out). • The queue is finite, limited to m calls

26. M/M/N/m system

27. M/M/N/m system: system with infinitequeue 1,2 B 1 m=0 m=1 0,8 m=2 0,6 m=5 0,4 m=10 0,2 m=# A 0 0 1 2 3 4 5 6 7 8 9 10 11 • Blocking/waiting probability in the system M/M/N/m Waiting probability as a function of the queue capacity in the M/M/5/m system. µ

28. M/G/1/∞ system – Assumptions • One server available for any call if it is not busy • Poisson arrival process with average intensity l • Any service time distribution with mean value 1/µ and variance σ2τ • Calls are waiting according to FIFO strategy (first in first out) • The queue is infinite. Carried traffic is equal to offered traffic

29. M/G/1/∞ system • Pollaczek-Khinchine’sformula • average number of calls in the system: • mean holding time in the system per call : variance of service time distribution.

30. M/G/1/∞ system • Pollaczek-Khinchine’s formula with residual service time: • Where: is the second moment of service time distribution

31. System M/D/1/∞ - Assumptions • One server available for any call if it is not busy,, • Poisson arrival process with average intensity l, • Constantservice time distribution with mean value 1/µ , • Calls are waiting according to FIFO strategy (first in first out). • Characteristics of the system M/D/1/∞ • Service time is constant, so its variance is equal to zero:

32. M/M/1/∞and M/D/1/∞systems comparison • Averagenumber of calls in the system

33. M/G/R PS system – Assumptions • Poisson arrival process with average intensityl • Anyservice time with mean value 1/μ • Availableresources are fairly divided between packet streams x offered to the system • Allthe offered streams are serviced quasi-simultaneously • Number of servers is equal to R

34. M/G/R PS system • M/G/R PS – special case of M/M/N/∞ system • A service of particular packet streams corresponds to the operation of mechanisms implemented in TCP protocol • Aspiration for assurance of equal access to a shared transmission channel • Convergence of models describing M/G/R PS and TCP • ModelM/G/R-PS is conventionally used for packet network dimensioning

35. System M/G/R PS • Number of servers • where: • V - capacity of the server (link) • Rmax-maximum bit rate of the traffic stream

36. System M/G/R PS system M/M/N/µ • Average time spent by a task (call) in the M/G/R PS system: • where: • fR- delay factor, • x - average length of task (call) x, for example, data file, • ρ- intensity of offered traffic to one server (from among R): • K- number of users.

37. System M/G/R PS • Average time spent by a task (call) in the M/G/R PS system: • where: • A- total offered traffic intensity: • E2,R(A) – Erlang’s C formula:

38. System M/G/R PS system M/M/1/µ • Average time spent by a task (call) in the M/G/R PS system with taking into account the delay in access link: • Delay in the access link: • where ρa is the traffic offered to access link with bit-rate equal to r:

39. M/G/R PS system dimensioning • Determination of the initial value of the link capacity V=r. • Determination of the transmission delay W(x)=f (ftotal). • Do the obtained delay values exceed required threshold ? • YES – increase capacity and go to step 2. • NO – required capacity has been reached. • Terminate calculation.

40. System M/M/1/m – bufferdimensioning system M/M/1/N-1 • The capacity m of the buffer for traffic with assumed QoS parameters can be determined on the basis of the acceptable level of loss packet probability E:

41. System M/M/1/m – buffer dimensioning • Approach 1 • The capacity m of the buffer for traffic with assumed QoS parameters can be determined on the basis of acceptable level of loss packet probability E: • Approach 2 • The capacity m of the buffer for traffic with assumed QoS parameters can be determined on the basis of average number of packet in queue Q:

42. Example – comparison of buffer dimensioning methods Model B m M/M/1/¥ (1) 7.06 10-9 24 M/M/1/¥ (2) 0.22 0.42 »1 M/M/1/N 3.70 10-9 24 • ATM links (150 Mbits/s) • traffic sources 1000 CBR sources (64 kbits/s) • required ATM packet intensity 166 700 packet/s • packet service time 2.830 µs • offered traffic intensity 0.472 Erl. • Determine required buffer capacity for