Queueing Fundamentals for Network Design Application ECE/CSC 777: Telecommunications Network Design Fall, 2013, Rudra Dutta

The Need for Models Designing or resource provisioning a system – “How much x of Resource X to put where?” – “So as to get y units of output behavior Y ” Prerequisite question to answer: – “What will be y for a given x ?” – “For any given x ?” Simulation by itself may not be helpful here – Okay if search space is small (few design choices) Analytical model is necessary – May not be very precise, still useful – May guide curve fits for simulation Copyright Rudra Dutta, NCSU, Fall, 2013

Statistical TDM Performance Bursty traffic, statistical TDM Usual M/M/1 assumptions – In reality, traffic process is heavier-tailed Delay is lower on average: “Statistical Multiplexing Gain” – But unpredictable for individual packet - prediction is statistical Link utilization /  Average Delay (ms) R Q

Copyright Rudra Dutta, NCSU, Fall, 2013 Blocking in Telephony Delay - very small and constant, operative quantity is blocking ratio Average call rate Average holding time Produce: offered traffic load or intensity X Q

Queueing Models Customer population Arrival process Service time distribution Buffer capacity Number of concurrent servers Queueing discipline Copyright Rudra Dutta, NCSU, Fall, 2013 Customer Arrivals Departures Buffer Server Infinite FIFO A/S/m/B/K/DsA/S/m/B/K/Ds Kendall Notation A/S/mA/S/m

Time Diagram Copyright Rudra Dutta, NCSU, Fall, 2013 nn  n+1  n+2 t n+1 t n+2 wnwn Service Queue snsn xnxn x n+1 x n+2 n-1

Little’s Law Mean number in system = arrival rate x mean response time – If T is large, arrivals = departures = N – Arrival rate = Total arrivals/Total time= N/T – Hatched areas = total time spent inside the system by all jobs = J – Mean time in the system= J/N – Mean Number in the system = J/T = J/N x N/T Copyright Rudra Dutta, NCSU, Fall, 2013

Memoryless Process Underlying process is such that nothing is known about time of next arrival except overall rate over long time – Next arrival time  time until which current random variable persists – How long we have been waiting does not tell us anything about when the wait will likely be over Does not change the distribution of the likelihood for future – How long the variable has survived does not change the likelihood distribution in the future conditioned by the present (past) Is (are) there such distributions? What? Copyright Rudra Dutta, NCSU, Fall, 2013

PDF of Termination over Time Bus leaves New York  Raleigh – Time 0 What is the distribution of likelihood (probability) of time of arrival? – Must sum (integrate) up to 1 Copyright Rudra Dutta, NCSU, Fall,

Termination Probability Distribution If the variable survives upto a particular time, this event conditions the probability of surviving for a further increment of time Changes the termination (equivalently, surviving) probability distribution from “here on” (in the subuniverse) Shape of remaining PDF remains, must scale to re- normalize Copyright Rudra Dutta, NCSU, Fall,

Memoryless Distribution Past does not change information about future – More precisely, the future, conditioned by the past, is the same as the original Must be infinitely long-tailed, must integrate to 1 e –t (more generally, e - t ) Rate is at time 0  (conditioned) rate is always Copyright Rudra Dutta, NCSU, Fall,

Markov Chain A discrete-state stochastic process in which probability of transitioning to another state only depends on present state – Memory of system is not zero, but limited Transition probabilities p ij, sojourn times q j Sojourn time must be exponentially or geometrically distributed (Markov property) Discrete time or continuous time – Discrete: transition probability at next tick (may be self-transitions) – Continuous: transition probability is exponential rate Semi-Markov: arbitrary sojourn time distribution Copyright Rudra Dutta, NCSU, Fall, (not how or when arrived at that state)

Markov Chain We will mostly be interested in Homogenous: unchanging transition probabilities Recurrent non-null: can eventually return to any state in finite time Aperiodic: return time to a state is possible at any time Irreducible: cannot be partitioned into non-communicating (steady-state) component chains – No unreachable or useless states Ergodic: aperiodic, recurrent, non-null Stationary: steady state exists Steady-state analysis: flow balance Copyright Rudra Dutta, NCSU, Fall,

Pure Birth Process Birth-death model – Population variation under births and deaths – Constant, independent rates – Time is continuous – chances of simultaneous events ignored – Only possible transitions: one more, or one less Pure birth – no death, continuously increasing population Copyright Rudra Dutta, NCSU, Fall,       

Pure Birth Process Rate of moving from any state to the next is – Over infinitesimal time  t, chance of transition is  t – Provided system in that state: un-condition with state probability at beginning of infinitesimal time Copyright Rudra Dutta, NCSU, Fall,        P k (t+  t) = P k (t) [1 -  t ] + P k-1 (t) [  t ] P k (t) : probability of population size k at time t  k P k (t) = 1, for any t d P k (t) _____ dtdt = - P k (t) + P k-1 (t)

Pure Birth Process Copyright Rudra Dutta, NCSU, Fall,        P k (t+  t) = P k (t) [1 -  t ] + P k-1 (t) [  t ] P k (t) : probability of population size k at time t  k P k (t) = 1, for any t d P k (t) _____ dtdt = - P k (t) + P k-1 (t) d P 0 (t) _____ dtdt = - P 0 (t) P 1 (0) = 1 P k (0) = 0, k ≠ 1  P 0 (t) = e - t  P 1 (t) = t e - t P k (t) = ( t) k e - t / k! Poisson distribution

Copyright Rudra Dutta, NCSU, Fall, 2013 Birth Death Process  μ      μμμμμμ p 2  + p 4 μ = p 1 = p  μ p k is the equilibrium state of P k (t) p 3 (  + μ ) p k = p  μ ( ) k  p k = 1 k=0 ∞  p 0 =  μ ( ) k 1 +  k=1 ∞ 1 = 1 – / 

The M/M/1 Queue Assumes Poisson arrival process, exponential service times, single server, FCFS service discipline, infinite capacity for storage Arrival rate: (e.g., packets / sec) – Inter-arrival times are exponentially distributed (and independent) with mean 1 / Service rate:  – Service times are exponentially distributed (and independent) with mean 1 /  System load / utilization:  –  must be strictly less than 1 for stability Copyright Rudra Dutta, NCSU, Fall, 2013

Performance Metrics N : Average number of customers in system, including any in service T : Average time spent in system p 0 = 1 – /  = 1 –  p k = (1 –  k N =  k p k k=0 ∞ N =  / (1 –  ) Little’s Law: N = T T = 1 / (  (1 –  ))  0 1  N =  / (1 –  ) 2 2 Copyright Rudra Dutta, NCSU, Fall, 2013

M/M/∞ System: Responsive server  μ      2μ3μ4μ5μ6μ7μ p k = p  (i+1)μ ( )  i=0 k-1 = p  μ ( ) k 1 k !  p 0 = e –  p k =  μ ( ) k e –  k ! N =  k p k k=0 ∞ N = /  Little’s Law: N = T T = 1 /  Naturally (consider second picture)

M/M/m/m Like M/M/∞/∞, but finite number of servers No buffering of waiting calls – blocked calls cleared Copyright Rudra Dutta, NCSU, Fall,  μ      2μ3μ4μ5μ6μ pkpk = p  μ ( ) k 1 k ! k ≤ m  p 0 = 1 / ---  μ ( ) k  k=0 m k ! p m =  μ ( ) k  k=0 m k ! ---  μ ( ) m m ! Erlang’s loss formula B (m,  ) Erlang’s B function E (A, N)

Summation Analytical models are necessary for network traffic arrival and service Model must represent – Uncertainty in exact time of traffic demand arrival, since this is not determined by network – Variation in effort required to serve traffic – too many factors, may represent by uncertainty Simple queuing considerations allow development of stochastic models Final result: models (formulae) that predict performance metrics (e.g. delay) – For any given amount of provisioned service resource – Under given traffic load (demand for service) Copyright Rudra Dutta, NCSU, Fall, 2013