CS 147: Computer Systems Performance Analysis
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1 CS 147: Computer Systems Performance Analysis Introduction to Queueing Theory CS 147: Computer Systems Performance Analysis Introduction to Queueing Theory 1 / 27
2 Overview Introduction and Terminology Poisson Distributions Overview Overview Introduction and Terminology Poisson Distributions Fundamental Results Stability Little s Law M/M/1 /B More General Queues Fundamental Results Stability Little s Law M/M/1 /B More General Queues 2 / 27
3 3 / 27 Introduction and Terminology What is a Queueing System? Introduction and Terminology What is a Queueing System? What is a Queueing System? A queueing system is any system in which things arrive, hang around for a while, and leave Examples A bank A freeway A computer) network A beehive The things that arrive and leave are customers or jobs Customers leave after receiving service Most queueing systems have surprise!) a queue that can store delay) customers awaiting service A queueing system is any system in which things arrive, hang around for a while, and leave Examples A bank A freeway A computer) network A beehive The things that arrive and leave are customers or jobs Customers leave after receiving service Most queueing systems have surprise!) a queue that can store delay) customers awaiting service
4 Introduction and Terminology Parameters of a Queueing System Arrival Process Injects customers into system Usually statistical Convenient to specify in terms of interarrival time distribution Most common is Poisson arrivals Introduction and Terminology Parameters of a Queueing System Parameters of a Queueing System Arrival Process Injects customers into system Usually statistical Convenient to specify in terms of interarrival time distribution Most common is Poisson arrivals Service Time Also statistical Number of Servers Often 1 System Capacity Equals number of servers plus queue capacity. Often assumed infinite for convenience Population Maximum number of customers. Often assumed infinite Service Discipline How next customer is chosen for service. Often FCFS or priority Service Time Also statistical Number of Servers Often 1 System Capacity Equals number of servers plus queue capacity. Often assumed infinite for convenience Population Maximum number of customers. Often assumed infinite Service Discipline How next customer is chosen for service. Often FCFS or priority 4 / 27
5 5 / 27 Introduction and Terminology Arrival and Service Distributions Introduction and Terminology Arrival and Service Distributions Arrival and Service Distributions Customer arrivals are random variables Next disk request from many processes Next packet hitting Google Next call to Chipotle Same is true for service times What distribution describes it? May be complicated fractal, Zipf) We often use Poisson for tractability Customer arrivals are random variables Next disk request from many processes Next packet hitting Google Next call to Chipotle Same is true for service times What distribution describes it? May be complicated fractal, Zipf) We often use Poisson for tractability
6 6 / 27 Introduction and Terminology Poisson Distributions The Poisson Distribution Probability of exactly k arrivals in 0, t) is P k t) = λt) k e λt /k! λ is arrival rate parameter More useful formulation is Poisson arrival distribution: PDF At) = P[next arrival takes time t] = 1 e λt pdf at) = λe λt Also known as exponential or memoryless distribution Mean = standard deviation = λ Poisson distribution is memoryless Assume P[arrival within 1 second] at time t0 = x Then P[arrival within 1 second] at time t1 > t 0 is also x I.e., no memory that time has passed Often true in real world E.g., when I go to Von s doesn t affect when you go Introduction and Terminology Poisson Distributions The Poisson Distribution The Poisson Distribution Probability of exactly k arrivals in 0, t) is Pkt) = λt) k e λt /k! λ is arrival rate parameter More useful formulation is Poisson arrival distribution: PDF At) = P[next arrival takes time t] = 1 e λt pdf at) = λe λt Also known as exponential or memoryless distribution Mean = standard deviation = λ Poisson distribution is memoryless Assume P[arrival within 1 second] at time t0 = x Then P[arrival within 1 second] at time t1 > t0 is also x I.e., no memory that time has passed Often true in real world E.g., when I go to Von s doesn t affect when you go
7 Introduction and Terminology Poisson Distributions Splitting and Merging Poisson Processes Introduction and Terminology Poisson Distributions Splitting and Merging Poisson Processes Splitting and Merging Poisson Processes Merging streams of Poisson events e.g., arrivals) is Poisson k λ = λi i=1 Splitting a Poisson stream randomly gives Poisson streams; if stream i has probability pi, then λi = piλ Merging streams of Poisson events e.g., arrivals) is Poisson λ = k i=1 λ i Splitting a Poisson stream randomly gives Poisson streams; if stream i has probability p i, then λ i = p i λ 7 / 27
8 8 / 27 Introduction and Terminology Poisson Distributions Kendall s Notation A/S/m/B/K/D defines a single) queueing system compactly: A Denotes arrival distribution, as follows: M Exponential Memoryless) E k Erlang with parameter k D Deterministic G Completely general very hard to analyze!) S Service distribution, same as arrival m Number of servers B System capacity; if omitted K Population size; if omitted D Service discipline, FCFS if omitted Introduction and Terminology Poisson Distributions Kendall s Notation Kendall s Notation A/S/m/B/K/D defines a single) queueing system compactly: A Denotes arrival distribution, as follows: M Exponential Memoryless) Ek Erlang with parameter k D Deterministic G Completely general very hard to analyze!) S Service distribution, same as arrival m Number of servers B System capacity; if omitted K Population size; if omitted D Service discipline, FCFS if omitted
9 Introduction and Terminology Poisson Distributions Examples of Kendall s Notation Introduction and Terminology Poisson Distributions Examples of Kendall s Notation Examples of Kendall s Notation D/D/1 Arrivals on clock tick, fixed service times, one server Memoryless arrivals, memoryless service, multiple servers good model of a bank) /m Customers go away rather than wait in line G/G/1 Modern disk drive D/D/1 Arrivals on clock tick, fixed service times, one server Memoryless arrivals, memoryless service, multiple servers good model of a bank) /m Customers go away rather than wait in line G/G/1 Modern disk drive 9 / 27
10 10 / 27 Introduction and Terminology Poisson Distributions Common Variables Introduction and Terminology Poisson Distributions Common Variables Common Variables τ Interarrival time. Usually varies per customer, e.g., τ1, τ2,... λ Mean arrival rate: 1/τ si Service time for job i, sometimes called xi µ Mean service rate per server, 1/s ρ Traffic intensity or system load = λ/mµ. This is the most important parameter in most queueing systems wi Waiting time, or time in queue: interval between arrival and beginning of service ri Response time = wi + si τ Interarrival time. Usually varies per customer, e.g., τ 1, τ 2,... λ Mean arrival rate: 1/τ s i Service time for job i, sometimes called x i µ Mean service rate per server, 1/s w i ρ Traffic intensity or system load = λ/mµ. This is the most important parameter in most queueing systems Waiting time, or time in queue: interval between arrival and beginning of service r i Response time = w i + s i
11 Fundamental Results Stability Stability Fundamental Results Stability Stability Stability A system is stable iff λ < mµ ρ < 1 Otherwise, system can t keep up and queue grows to Exception: in D/D/m, ρ = 1 is OK A system is stable iff λ < mµ ρ < 1 Otherwise, system can t keep up and queue grows to Exception: in D/D/m, ρ = 1 is OK 11 / 27
12 12 / 27 Fundamental Results Little s Law Little s Law Let n = Number of jobs in system Fundamental Results Little s Law Little s Law Little s Law Let n = Number of jobs in system Then n = λr Likewise, if nq = Number of jobs in queue, then nq = λw True regardless of distributions, queueing disciplines, etc., as long as system is in equilibrium May seem obvious: If ten people are ahead of you in line, and each takes about 1 minute for service, you re going to be stuck there for 10 minutes Not proved until 1961 Often useful for calculating queue lengths: Packet takes 2s to arrive, you re sending 100 pps Mean queue length = 100 pkt/s 2s = 200 pkts Then n = λr Likewise, if n q = Number of jobs in queue, then n q = λw True regardless of distributions, queueing disciplines, etc., as long as system is in equilibrium May seem obvious: If ten people are ahead of you in line, and each takes about 1 minute for service, you re going to be stuck there for 10 minutes Not proved until 1961 Often useful for calculating queue lengths: Packet takes 2s to arrive, you re sending 100 pps Mean queue length = 100 pkt/s 2s = 200 pkts
13 Fundamental Results Little s Law Deriving Little s Law Define arrt) = # of arrivals in interval0, t) Fundamental Results Little s Law Deriving Little s Law Deriving Little s Law Define arrt) = # of arrivals in interval0, t) Define dept) = # of departures in interval0, t) Clearly, Nt) = # in system at timet = arrt) dept) Area between curves = spentt) = total time spent in system by all customers measured in customer-seconds) 10 Nt) arrt) 2 dept) Define dept) = # of departures in interval0, t) Clearly, Nt) = # in system at timet = arrt) dept) Area between curves = spentt) = total time spent in system by all customers measured in customer-seconds) arrt) dept) Nt) 13 / 27
14 14 / 27 Fundamental Results Little s Law Deriving Little s Law continued) Define average arrival rate during interval t, in customers/second, as λ t = arrt)/t Define T t as system time/customer, averaged over all customers in 0, t) Since spentt) = accumulated customer-seconds, divide by arrivals up to that point to get T t = spentt)/arrt) Mean tasks in system over 0, t) is accumulated customer-seconds divided by seconds: Mean-tasks t = spentt)/t Above three equations give us: Fundamental Results Little s Law Deriving Little s Law continued) Deriving Little s Law continued) Define average arrival rate during interval t, in customers/second, as λt = arrt)/t Define Tt as system time/customer, averaged over all customers in 0, t) Since spentt) = accumulated customer-seconds, divide by arrivals up to that point to get Tt = spentt)/arrt) Mean tasks in system over 0, t) is accumulated customer-seconds divided by seconds: Mean-taskst = spentt)/t Above three equations give us: Mean-taskst = spentt)/t = Ttarrt)/t = λttt Mean-tasks t = spentt)/t = T t arrt)/t = λ t T t
15 Fundamental Results Little s Law Deriving Little s Law continued) Fundamental Results Little s Law Deriving Little s Law continued) Deriving Little s Law continued) We ve shown that Mean-taskst = λttt Assuming limits of λt and Tt exist, limit of mean-taskst also exists and gives Little s result: Mean tasks in system = arrival rate mean time in system We ve shown that Mean-tasks t = λ t T t Assuming limits of λ t and T t exist, limit of mean-tasks t also exists and gives Little s result: Mean tasks in system = arrival rate mean time in system 15 / 27
16 16 / 27 The M/M/1 Queue M/M/1 Remember this one if you don t remember anything else Assumptions are sometimes realistic, sometimes not Never infinite customers or capacity Service times aren t truly Poisson Interarrival times more likely to be Poisson Still provides surprisingly good analysis M/M/1 s characteristics are clue to many other queues Primary results in equilibrium): Mean number in system n = ρ/1 ρ) Mean time in system r = 1/µ)/1 ρ) = 1/µ1 ρ) = s/1 ρ) M/M/1 The M/M/1 Queue The M/M/1 Queue Nearly all useful results in queueing theory apply only to systems in equilibrium. Remember this one if you don t remember anything else Assumptions are sometimes realistic, sometimes not Never infinite customers or capacity Service times aren t truly Poisson Interarrival times more likely to be Poisson Still provides surprisingly good analysis M/M/1 s characteristics are clue to many other queues Primary results in equilibrium): Mean number in system n = ρ/1 ρ) Mean time in system r = 1/µ)/1 ρ) = 1/µ1 ρ) = s/1 ρ)
17 17 / 27 The Nastiness of High Load Mean Number in System M/M/ ρ M/M/1 The Nastiness of High Load The Nastiness of High Load Mean Number in System ρ The system breaks down completely at ρ > The reason for the breakdown is variance: at high load, a burst fills the queue and it takes a long time to drain, giving plenty of time for another burst to arrive.
18 18 / 27 M/M/1 More M/M/1 Results Variance is ρ/1 ρ) 2, so standard deviation is ρ/1 ρ) q-percentile of time in system is r ln[100/100 q)] 90 th percentile is 2.3r M/M/1 More M/M/1 Results More M/M/1 Results Variance is ρ/1 ρ) 2, so standard deviation is ρ/1 ρ) q-percentile of time in system is r ln[100/100 q)] 90 th percentile is 2.3r Mean waiting time is w = 1 ρ µ 1 ρ q-percentile of waiting time is max 0, wρ ) ln[100ρ/100 q)] Mean jobs served in a busy period: 1/1 ρ) Probability of n jobs in system pn = 1 ρ)ρ n Probability of > n jobs in system: ρ n Mean waiting time is w = 1 µ 1 ρ q-percentile of waiting time is max 0, wρ ) ln[100ρ/100 q)] ρ Mean jobs served in a busy period: 1/1 ρ) Probability of n jobs in system p n = 1 ρ)ρ n Probability of > n jobs in system: ρ n
19 M/M/1 M/M/1 Example M/M/1 M/M/1 Example M/M/1 Example Web server gets 1200 requests/hour w/ Poisson arrivals Typical request takes 1s to serve ρ = 1200/3600 = 0.33 Mean requests in service = 0.33/0.67 = 0.5 Mean response time r = 1/1)/1 0.33) = 1.5s 90 th percentile response time = 3.4s Web server gets 1200 requests/hour w/ Poisson arrivals This slide has animations. Typical request takes 1s to serve ρ = 1200/3600 = 0.33 Mean requests in service = 0.33/0.67 = 0.5 Mean response time r = 1/1)/1 0.33) = 1.5s 90 th percentile response time = 3.4s 19 / 27
20 M/M/1 M/M/1 Example M/M/1 M/M/1 Example M/M/1 Example Web server gets 1200 requests/hour w/ Poisson arrivals Typical request takes 1s to serve ρ = 1200/3600 = 0.33 Mean requests in service = 0.33/0.67 = 0.5 Mean response time r = 1/1)/1 0.33) = 1.5s 90 th percentile response time = 3.4s But if Slashdot hits... Web server gets 1200 requests/hour w/ Poisson arrivals This slide has animations. Typical request takes 1s to serve ρ = 1200/3600 = 0.33 Mean requests in service = 0.33/0.67 = 0.5 Mean response time r = 1/1)/1 0.33) = 1.5s 90 th percentile response time = 3.4s But if Slashdot hits / 27
21 M/M/1 M/M/1 Example cont d) M/M/1 M/M/1 Example cont d) M/M/1 Example cont d) Suppose Slashdot raises request rate to 3500/hr Now ρ = 3500/3600 = Mean requests in service = 0.972/ ) = 34.7 r = 1/0.028 = 35.7 seconds 90 th percentile response time = 82.8s Suppose Slashdot raises request rate to 3500/hr This slide has animations. Now ρ = 3500/3600 = Mean requests in service = 0.972/ ) = 34.7 r = 1/0.028 = 35.7 seconds 90 th percentile response time = 82.8s 20 / 27
22 M/M/1 M/M/1 Example cont d) M/M/1 M/M/1 Example cont d) M/M/1 Example cont d) Suppose Slashdot raises request rate to 3500/hr Now ρ = 3500/3600 = Mean requests in service = 0.972/ ) = 34.7 r = 1/0.028 = 35.7 seconds 90 th percentile response time = 82.8s And don t even think about 4000 requests/hr Suppose Slashdot raises request rate to 3500/hr This slide has animations. Now ρ = 3500/3600 = Mean requests in service = 0.972/ ) = 34.7 r = 1/0.028 = 35.7 seconds 90 th percentile response time = 82.8s And don t even think about 4000 requests/hr 20 / 27
23 21 / 27 Multiple servers, one queue ρ = λ/mµ) We ll need probability of empty system: For m = 1, ϱ = ρ Multiple servers, one queue ρ = λ/mµ) We ll need probability of empty system: p0 = Probability of queueing: 1 mρ) m m 1 m!1 ρ) + ϱ = P m jobs) = k=0 mρ) k k! mρ)m m!1 ρ) p0 p 0 = 1 mρ) m m 1 m!1 ρ) + k=0 mρ) k k! Probability of queueing: ϱ = P m jobs) = mρ)m m!1 ρ) p 0
24 22 / 27 cont d) cont d) cont d) Mean jobs in system: n = mρ + ρϱ/1 ρ) Mean time in system: r = 1 µ 1 + ) ϱ m1 ρ) Mean waiting time: w = ϱ/[mµ1 ρ)] q-percentile of waiting time: max 0, w ϱ ) 100ϱ ln 100 q Mean jobs in system: n = mρ + ρϱ/1 ρ) Mean time in system: r = 1 µ ) ϱ 1 + m1 ρ) Mean waiting time: w = ϱ/[mµ1 ρ)] q-percentile of waiting time: max 0, w ϱ ) 100ϱ ln 100 q
25 23 / 27 k=0 mρ) k k! m M/M/1 vs. For m separate M/M/1 queues, each queue sees arrival rate of λ M/M/1 = λ/m But ρ is unchanged r m M/M/1 = 1 1?> ) µ 1 ρ) r = 1 µ 1 + ϱ m1 ρ) m M/M/1 vs. This slide has animations. m M/M/1 vs. For m separate M/M/1 queues, each queue sees arrival rate of λ M/M/1 = λ/m But ρ is unchanged r m M/M/1 = 1 1?> ) µ 1 ρ) r = 1 µ 1 + ϱ m1 ρ) 1? > 1 ρ + ϱ m ρ >? mρ) m p0 m!m1 ρ) 1 >? p0 mρ)m 1 m!1 ρ) 1? > 1 mρ) m m!1 ρ) + m 1 ) mρ) m 1 m!1 ρ) 1? > 1 ρ + ϱ m ρ? > p 0 mρ) m m!m1 ρ) 1 >? mρ) p m 1 0 m!1 ρ) 1? > 1 mρ) m m!1 ρ) + m 1 mρ) k k=0 k! ) mρ) m 1 m!1 ρ)
26 23 / 27 k=0 mρ) k k! m M/M/1 vs. For m separate M/M/1 queues, each queue sees arrival rate of λ M/M/1 = λ/m But ρ is unchanged r m M/M/1 = 1 1?> ) µ 1 ρ) r = 1 µ 1 + ϱ m1 ρ) m M/M/1 vs. This slide has animations. m M/M/1 vs. For m separate M/M/1 queues, each queue sees arrival rate of λ M/M/1 = λ/m But ρ is unchanged r m M/M/1 = 1 1?> ) µ 1 ρ) r = 1 µ 1 + ϱ m1 ρ) 1? > 1 ρ + ϱ m ρ >? mρ) m p0 m!m1 ρ) 1 >? p0 mρ)m 1 m!1 ρ) 1? > 1 mρ) m m!1 ρ) + m 1 mρ)m m!1 ρ) + m 1 k=0 ) mρ) m 1 m!1 ρ) mρ) k k! > mρ)m 1 m!1 ρ) 1? > 1 ρ + ϱ m ρ? > p 0 mρ) m m!m1 ρ) 1 >? mρ) p m 1 0 m!1 ρ) 1? > 1 mρ) m m!1 ρ) + m 1 mρ) k k=0 k! ) mρ) m 1 m!1 ρ) mρ)m m!1 ρ) + m 1 mρ) k k=0 k! > mρ)m 1 m!1 ρ)
27 23 / 27 k=0 mρ) k k! m M/M/1 vs. For m separate M/M/1 queues, each queue sees arrival rate of λ M/M/1 = λ/m But ρ is unchanged r m M/M/1 = 1 1?> ) µ 1 ρ) r = 1 µ 1 + ϱ m1 ρ) m M/M/1 vs. This slide has animations. m M/M/1 vs. For m separate M/M/1 queues, each queue sees arrival rate of λ M/M/1 = λ/m But ρ is unchanged r m M/M/1 = 1 1?> ) µ 1 ρ) r = 1 µ 1 + ϱ m1 ρ) 1? > 1 ρ + ϱ m ρ >? mρ) m p0 m!m1 ρ) 1 >? p0 mρ)m 1 m!1 ρ) 1? > 1 mρ) m m!1 ρ) + m 1 mρ)m m!1 ρ) + m 1 k=0 ) mρ) m 1 m!1 ρ) mρ) k k! > mρ)m 1 m!1 ρ) 1? > 1 ρ + ϱ m ρ? > p 0 mρ) m m!m1 ρ) 1 >? mρ) p m 1 0 m!1 ρ) 1? > 1 mρ) m m!1 ρ) + m 1 mρ) k k=0 k! ) mρ) m 1 m!1 ρ) mρ)m m!1 ρ) + m 1 mρ) k k=0 k! > mρ)m 1 m!1 ρ)
28 23 / 27 k=0 mρ) k k! m M/M/1 vs. For m separate M/M/1 queues, each queue sees arrival rate of λ M/M/1 = λ/m But ρ is unchanged r m M/M/1 = 1 1?> ) µ 1 ρ) r = 1 µ 1 + ϱ m1 ρ) m M/M/1 vs. This slide has animations. m M/M/1 vs. For m separate M/M/1 queues, each queue sees arrival rate of λ M/M/1 = λ/m But ρ is unchanged r m M/M/1 = 1 1?> ) µ 1 ρ) r = 1 µ 1 + ϱ m1 ρ) 1? > 1 ρ + ϱ m ρ >? mρ) m p0 m!m1 ρ) 1 >? p0 mρ)m 1 m!1 ρ) 1? > 1 mρ) m m!1 ρ) + m 1 mρ)m m!1 ρ) + m 1 k=0 ) mρ) m 1 m!1 ρ) mρ) k k! > mρ)m 1 m!1 ρ) 1? > 1 ρ + ϱ m ρ? > p 0 mρ) m m!m1 ρ) 1 >? mρ) p m 1 0 m!1 ρ) 1? > 1 mρ) m m!1 ρ) + m 1 mρ) k k=0 k! ) mρ) m 1 m!1 ρ) mρ)m m!1 ρ) + m 1 mρ) k k=0 k! > mρ)m 1 m!1 ρ)
29 24 / 27 mρ)m k=0 mρ) k k! 4! Running Some Numbers Assume 5 servers, ρ = 0.5, µ = 1 Running Some Numbers Running Some Numbers Assume 5 servers, ρ = 0.5, µ = 1 Then r m M/M/1 = 1/1 ρ) = 2 2.5) ϱ = m!1 ρ) p0 = 5!0.5) p0 = 60 p0 = 1.63p0 1 1 p0 = = mρ) m m!1 ρ) + m ! 1 p0 = = = 0.08 So ϱ = ) = 0.13 And r m/m/m = 1 + ϱ m1 ρ) = ) = = 1.05 In terms of previous slide s inequality, = > 5!0.5) = = 0.65 Then r m M/M/1 = 1/1 ρ) = 2 ϱ = mρ)m m!1 ρ) p 0 = 2.5)5 5!0.5) p 0 = p 0 = 1.63p 0 1 p 0 = mρ) m m!1 ρ) + m 1 mρ) k k=0 k! 1 = ! 4! 1 p 0 = = = 0.08 So ϱ = ) = 0.13 And r m/m/m = 1 + ϱ m1 ρ) = ) = = 1.05 In terms of previous slide s inequality, = > 5!0.5) = = 0.65
30 m M/M/1 vs. cont d) m M/M/1 vs. cont d) m M/M/1 vs. cont d) A similar result holds for variance Conclusion: single queue, multiple server is always better than one queue per server Question 1: When is this false? hint: multiple cores) Question 2: Why do so many movie theaters have multiple lines for popcorn? A similar result holds for variance Conclusion: single queue, multiple server is always better than one queue per server Question 1: When is this false? hint: multiple cores) Question 2: Why do so many movie theaters have multiple lines for popcorn? 25 / 27
31 26 / 27 /B /B /B /B /B Real systems have finite capacity Previous analysis applies only under light loads relative to capacity) Considering limit has several effects: Lost jobs obviously) Loss rate pb becomes important parameter Mean response time drops compared to / Why?) Real systems have finite capacity Previous analysis applies only under light loads relative to capacity) Considering limit has several effects: Lost jobs obviously) Loss rate p B becomes important parameter Mean response time drops compared to / Why?)
32 27 / 27 More General Queues Extending the Results More General Queues Extending the Results Extending the Results Unsurprisingly, generality equates to mathematical) complexity Many special cases have been analyzed e.g., Erlang distributions) Little s Law always applies Important cases: M/G/1 M/D/1 G/G/m but mostly intractable) Unsurprisingly, generality equates to mathematical) complexity Many special cases have been analyzed e.g., Erlang distributions) Little s Law always applies Important cases: M/G/1 M/D/1 G/G/m but mostly intractable)
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