Maciej Stasiak, Mariusz Głąbowski Arkadiusz Wiśniewski, Piotr Zwierzykowski Model of the Nodes in the Packet Network Chapter 10

Queuing system 2

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

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

Kendall’s notation (3) Service strategy (example of standard strategies) o FCFS: First Come – First Served, i.e. ordered queue, waiting calls are serviced in successive order; o 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; o 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; 5

Little’s Theorem Basic system parameters: o L the average number of calls in the system, o Wmean holding time in the system per call, o Qthe average number of calls in the queue, o T mean holding time in the queue per call. 6 L =  W Q = T the average number of calls in the queue = X the average call intensity mean holding time in the queue per call

Little’s Theorem A(t) number of arrivals at the moment t, B(t) number of calls outgoing from the system 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. 7 Arrival and departure process in the queuing system

Little’s Theorem 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: 8      0 )( )( 1 WttdttZL i i i i

One server delay system with infinite queue M/M/1/∞ One server available for any call if it is not busy, Poisson arrival process with average intensity, o exponential service time with mean value 1/μ, o 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. 9 QUEUE = ∞ output stream input stream (λ) SERVER μ

M/M/1/ ∞ system State transition diagram 10  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 served and n-1 calls are waiting in the queue. ·...,

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

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

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

M/M/1/ ∞ system - characteristics Average number of calls in the system – formula derivation: 14

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

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

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

M/M/N/∞ system N servers available for any call if are not busy, Poisson arrival process with average intensity, o exponential service time with mean value 1/μ, o 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. 18

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

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

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

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

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): 23

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

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

M/M/N/m system 26

M/M/N/m system: system with infinite queue Blocking/waiting probability in the system M/M/N/m 27 Waiting probability as a function of the queue capacity in the M/M/5/m system. 

M/G/1/∞ system – Assumptions One server available for any call if it is not busy Poisson arrival process with average intensity Any service time distribution with mean value 1/µ and variance σ 2 τ Calls are waiting according to FIFO strategy (first in first out) The queue is infinite. Carried traffic is equal to offered traffic 28

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

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

System M/D/1/∞ - Assumptions One server available for any call if it is not busy,, Poisson arrival process with average intensity, Constant service 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/∞ o Service time is constant, so its variance is equal to zero: 31

M/M/1/∞ and M/D/1/∞ systems comparison Average number of calls in the system 32

M/G/R PS system – Assumptions Poisson arrival process with average intensity  Any service time with mean value 1/μ Available resources are fairly divided between packet streams x offered to the system All the offered streams are serviced quasi- simultaneously Number of servers is equal to R 33

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 Model M/G/R-PS is conventionally used for packet network dimensioning 34

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

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

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

System M/G/R PS 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: o where ρ a is the traffic offered to access link with bit-rate equal to r: 38 system M/M/1/ 

M/G/R PS system dimensioning 1.Determination of the initial value of the link capacity V=r. 2.Determination of the transmission delay W(x)=f (f total ). 3.Do the obtained delay values exceed required threshold ? 1.YES – increase capacity and go to step 2. 2.NO – required capacity has been reached. 4.Terminate calculation. 39

System M/M/1/m – buffer dimensioning 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: 40 system M/M/1/ 

System M/M/1/m – buffer dimensioning Approach 1 o 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 o 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: 41

Example – comparison of buffer dimensioning methods ATM links (150 Mbits/s) o traffic sources1000 CBR sources (64 kbits/s) o required ATM packet intensity packet/s o packet service time2.830 µs o offered traffic intensity0.472 Erl. o Determine required buffer capacity for M/M/1/N 0.42  M/M/1/  (2) M/M/1/  (1) mBModel