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Model of the Nodes in the Packet Network Chapter 10

Model of the Nodes in the Packet Network Chapter 10. Queuing system. Kendall ’s notation (1). Classification of queuing system depends on: Structure: number of servers Arrival stream: interarrival time distribution Service stream: service time distribution

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Model of the Nodes in the Packet Network Chapter 10

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  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

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