Queueing Theory 14-740: Fundamentals of Computer Networks Bill Nace Material from Computer Networking: A Top Down Approach, J.F. Kurose and K.W. Ross
Administrivia • Quiz #1 is next lesson, on Canvas • Study Guide is on the website • Covers all material so far, including today • Includes readings (text and papers) • Closed book, closed notes • No calculator needed • HW #1 due the next Wed (6 Oct) • Lab #1 is now posted 2
Quiz Details • Equation sheet on the website • 15 Questions • 2 Medium Answer • 5 T/F • 1 Queueing • 3 Multiple Answer Theory • 4 Short Answer 3
traceroute • QT Overview • Performance Evaluation • Little’s Law • Rate Transition Diagrams • M/M/1 Systems • M/M/c Systems • Examples 4
♫ Queueing Theory • An analytic tool to make performance statements about queueing processes Waiting Served Customers Customers (leaving) Arriving Customers Service Facility Customers being Served Discouraged Customers (leaving) • Very applicable: retail, manufacturing, ... • Network uses: routers, transport, ... 5
Queueing System • Describe via six characteristics • Arrival pattern • Service pattern • Queue discipline • System capacity • # of service channels • # of service stages 6
#1: Arrival Pattern • Often, arrivals are stochastic • Need to know the PDF of interarrival times • Sometimes, arrive in batches or in bulk • Need the PDF of batch size • A stationary arrival pattern doesn’t change with time 7
Impatient Customers • Customer reaction is sometimes impatient • Balk is when a customer refuses to enter • Renege is when customer leaves • Customer may jockey for position by switching input queues • Network packets rarely do any of these 8
#2: Service Pattern • Similar to arrival patterns, service patterns are often stochastic • Described by a PDF of customer service times • A state-dependent server changes based on the number of customers waiting • Server may get flustered or work faster • Service can be stationary or nonstationary 9
#3: Queue Discipline • The manner in which customers are chosen from the queue for service • First Come First Served (FCFS) • Last Come First Served • Useful for stack based or inventory systems • Random Service Selection (RSS) • Priority Schemes 10
#4: System Capacity • Is there a physical limitation to the number of customers in the queue? • Common in real life • Bu ff er memory in a router • # of chairs for waiting at barber shop • Often ignored to simplify the analysis 11
♫ ♫ ♫ #5: Multiple Service Channels • Adding servers increases capacity • How do customers wait? • Single queue can feed many servers • Each server may have its own queue Service Service Facility Facility Service Service Facility Facility Service Service Facility Facility 12
#6: Stages of Service • In a multistage queueing system, customers exit one service only to start waiting for another service • Doctor’s O ffi ce: check-in, history, exam, tests, re-exam, payment • Some systems allow feedback (recycling parts is common in manufacturing) 13
Kendall53’s A/B/X/Y/Z Notation Characteristic Symbol Explanation M Exponential Interarrival-time D Deterministic distribution (A) E k Erlang type k H k k exponentials Service-time PH Phase Type distribution (B) G General 1, 2, 3... , ∞ # parallel servers (X) 1, 2, 3... , ∞ Max capacity ( Y ) FCFS First come, first served LCFS Last come, first served Queue Discipline ( Z ) RSS Random Selection for Service PR Priority GD General Discipline 14
Common Queueing Systems • G/G/1 ➙ General, single server system • G/G/c ➙ General, multi-server system • G/G/ ∞ ➙ General, self-serve system • M/M/1 ➙ Markovian, single server • M/M/c ➙ Markovian, multi-server • M/M/c/k ➙ Multi-server, with limited size • Infinite system size and FCFS discipline are always assumed as the defaults 15
traceroute • QT Overview • Performance Evaluation • Little’s Law • Rate Transition Diagrams • M/M/1 Systems • M/M/c Systems • Examples 16
Parameters • λ is the average arrival rate • # customers or packets per second • μ is the average service rate • # packets served per second • c is number of servers 17
Tra ffi c intensity • ρ ≡ λ / c μ • A measure of tra ffi c congestion in the servers • When ρ > 1, average # of arrivals exceeds service capability ➙ bad • When ρ = 1, randomness prevents queues from emptying (unless perfectly scheduled deterministic arrivals) ➙ bad • Steady state only when ρ < 1 18
# of customers • N(t) is number of customers in the system • Sum of N q (t) and N s (t) , for queues and service • Expected number in system ∞ X L = E [ N ] = np n n =0 • Expected number in queue ∞ X L q = E [ N q ] = ( n − c ) p n n = c +1 • Where p n is Pr{N=n}, probability of a particular number n customers in the system 19
Time • TI is interarrival time (time between successive arrivals) • T q is time spent in queue • S is service time Arrival of Arrival of Departure of Departure of Customer n Customer n+1 Customer n-1 Customer n • T is total time TI(n) (n) S(n) Tq • T = T q + S (n+1) Tq 20
Little’s Law • Relationship between # customers and the waiting time • W is the mean waiting time in the system • W = E[ T ] and W q = E[ T q ] • Little’s Law: L = λ W ( and L q = λ W q ) • Especially powerful when combined with E[ T ] = E[ T q ] + E[ S ] to get W = W q + 1/ μ • Given λ , μ and any 1 of {W, W q , L, L q } can get rest 21
Further Results • What is E[ N s ]? • L - L q = λ (W - W q ) = λ (1/ μ ) = λ / μ • r ≡ λ / μ • For c=1, r = ρ and ∞ ∞ ∞ X X X L − L q = ( n − 1) p n = p n = 1 − p 0 np n − n =1 n =1 n =1 22
Busy Probability • For a G/G/c system, P b is the probability of a given server being busy • At steady state, E[ N s ] = r • Servers are identical, so • E[N a server ] = r/c = P b • Recall that ρ = λ / c μ and r = λ / μ • Therefore, P b = ρ 23
G/G/c Summary ρ = λ /c μ Traffic intensity L = λ W Little’s Law L q = λ W q Little’s Law W = W q +1/ μ P b = λ /c μ = ρ Busy probability for an arbitrary server r = λ / μ Expected number of customers in service L = L q + r p 0 = 1 - ρ G/G/1 empty-system probability G/G/1 combined result L = L q + (1-p 0 ) 24
traceroute • QT Overview • Performance Evaluation • Little’s Law • Rate Transition Diagrams • M/M/1 Systems • M/M/c Systems • Examples 25
Birth-death process • A type of continuous-time Markov chain • System modeled as a set of states • Transitions occur between adjacent states • When in state n , an arrival moves the system to state n +1 • When in state n > 0, a departure moves the system to state n -1 26
Rate Transition Diagram λ 0 λ 1 λ 2 λ 3 λ 4 0 1 2 3 4 μ 1 μ 2 μ 3 μ 4 μ 5 • Describes the number of customers in the system • Customers arrival time is an exponential random variable with rate λ n • Likewise, departure is random with rate μ n 27
Flow-balance Equations λ 0 λ 1 λ 2 λ 3 λ 4 0 1 2 3 4 μ 1 μ 2 μ 3 μ 4 μ 5 • p n is the probability of being in state n • The system is in steady-state, therefore: ( λ n + μ n )p n = λ n-1 p n-1 + μ n+1 p n+1 for n ≥ 1 λ 0 p 0 = μ 1 p 1 28
• My apologies for the derivation! • Rewriting the flow-balance equations: λ n + µ n p n − λ n − 1 = p n +1 p n − 1 µ n +1 µ n +1 λ 0 = p 1 p 0 µ 1 • Do some inductive reasoning: λ 1 + µ 1 p 1 − λ 0 = p 2 p 0 µ 2 µ 2 λ 1 + µ 1 λ 0 p 0 − λ 0 = p 0 µ 2 µ 1 µ 2 Which leads to: λ 1 λ 0 = p 0 µ 2 µ 1 • Similarly: λ n − 1 λ n − 2 · · · λ 0 p n p 0 = µ n µ n − 1 · · · µ 1 p 3 = λ 2 λ 1 λ 0 n λ i − 1 p 0 Y µ 3 µ 2 µ 1 p 0 = µ i i =1 29
traceroute • QT Overview • Performance Evaluation • Little’s Law • Rate Transition Diagrams • M/M/1 Systems • M/M/c Systems • Examples 30
M/M/1 Systems • Exponentially distributed because TI ( t ) = λ e − λ t Interarrival time S ( t ) = µe − µt Service time • Independent of state of the system λ λ λ λ λ 0 1 2 3 4 μ μ μ μ μ 31
M/M/1 Flow Balance Eqns • Rewrite the G/G/c eqns with λ and μ • λ n = λ and μ n = μ for all n n ◆ n ✓ λ ◆ ✓ λ Y p n = p 0 = p 0 µ µ i =1 • Now what? • We need to know what p 0 is! 32
Probabilities must add to 1 ∞ ∞ ∞ ◆ n ✓ λ X X X ρ n 1 = p n = = p 0 p 0 µ n =0 n =0 n =0 • (Remember ρ = λ /c μ and c=1 server) ∞ X ρ n converges if ρ < 1 n =0 ∞ 1 X ρ n = 1 − ρ n =0 p 0 1 = 1 − ρ = 1 − ρ p 0 33
Plug p 0 into Flow-Balance = 1 − ρ gets plugged into p 0 ◆ n ✓ λ = to produce p n p 0 µ = (1 − ρ ) ρ n p n • This final equation is incredibly useful, as it allows us to determine probabilities of system state, given only λ and μ • Also allows us to generate measures of e ff ectiveness 34
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