Transcript Performance Engineering
Performance Engineering
QUEUEING
Prof. Jerry Breecher Queueing Models
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WHAT WE ARE DOING HERE
We are about to embark on: Queueing Lingo all the definitions you’ll ever need!
Queueing Analytical Models especially the Single Queue in DETAIL Operational Laws how to solve a number of complex problems using the equations we already know and love. Works especially well for open models.
drawing conclusions about limits - what to do when we can’t solve the math exactly.
Mean Value Analysis Using the equations in an iterative fashion to solve closed models.
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Queueing Models
This section is about being able to describe the behavior of queues. Queues are certainly a prevalent object in computer systems, and our goal here is to write the equations that describe them.
The language we’ll use here is mathematical, but nothing really more complicated than algebra.
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Queueing Lingo
Goals:
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To understand the random/statistical nature of computer data. We will emphasize the non-deterministic.
To understand distributions for simulation purposes.
To impress your friends.
For an observation, two things matter:
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The value measured.
When measured.
The occurrence of an event can give us the "when".
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Queueing Lingo
A STOCHASTIC PROCESS
"random, statistical".
is a mechanism that produces a collection of measurements which all occur, randomly, in the same range of values.
applies to the VALUE measured for an observation.
It The dictionary says, Stochastic processes are well behaved phenomena which don't do things which are unpredictable or unplanned for.
Examples:
Throwing 2 dice always gives numbers in the range 2 - 12.
Actions of people are unpredictable ( unless the range of values is made very large.) Someone can always respond in a way you haven't predicted.
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Queueing Lingo
THE POISSON PROCESS
applies to WHEN an observation is made.
It looks random; the arrival points are uniformly distributed across a time interval.
Poisson processes can be defined by:
Event counting
The distribution of the number of events occurring in a particular time is a Poisson distribution.
Time between events
The distribution of times between event occurrences is exponential.
Example:
Show how a random "look" leads to an exponential distribution. See the next page for a picture of these distributions.
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F(t) = exp(-t)
Queueing Lingo
Exponential Curve 1 0.9
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F(t)
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t 2 2.5
3 This is a simple exponential curve. What properties can you identify from it?
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Queueing Lingo
F(k) = ( 5 / k! ) exp( -5 ) Poisson Probability Density Function 0.2
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0 0 2 4 6 k 8 Example of the Poisson Probability Density Function.
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Lab - You Get To See It Happen Before Your Very Eyes!
This is a group lab designed to demonstrate that the number of random events in a particular interval is a Poisson distribution, and that the distribution of spaces between events is exponential.
Open your textbook 20 times. Each time, write down the number of the left hand page ( mod 100 ) beside the appropriate sample in column I.
Count how many of the segments in Column I have 0 samples in them. Put this number in the first row of Column II. Repeat for 1, 2, 3, ... samples.
For each of the samples in Column I, determine the distance or separation between that sample and the next higher sample. If the the distance is 0 ( the two samples are the same ) put a tick in the first row of column 3, if the separation is 3, tick the interval = 3 row, etc.
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Lab - You Get To See It Happen Before Your Very Eyes!
Column I Numbers falling in each segment.
Column II Number of segments containing N samples Column III Number of instances of intervals between samples.
Interval Segment How Many Samples N Number of Segments Number of Instances 0 0
0 – 9 10 – 19 20 - 29
1 2 2 4 6
30 - 39
3 8
40 - 49
4 10
50 - 59 60 - 69
5 6 12 14
70 - 79
7 16
80 - 89 90 - 99
8 9 18
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Lab - You Get To See It Happen Before Your Very Eyes!
// Generate Poisson.
// This program generates random numbers, puts them into buckets, // and calculates the distance between numbers.
// // Inputs: // N - number of samples to generate #include for ( Index = 0; Index < MAX_BUCKETS; Index++ ) Bucket[Index] = 0; if ( argc <2 ) { printf( "Usage: GeneratePoisson Code is here so it doesn’t get lost!! The results of running this code can be seen here Experimental Results This is the result of running 4 experiments of 20 samples each. The results aren’t pretty!! Perhaps by running the tests many times, everything would look nice and smooth. You can see the results you would expect, but there’s lots of jitter. Queueing Models • • Examples: Suppose that a piece of software has an expected lifetime of 5 years, and that the average bug rate for this type of product is one bug/year. What is the bug expectation rate per year at the start of the five years, assuming this code is "average"? After two years, four bugs have been found. considered "average". remaining three years? The code is still How many bugs/year can be expected for the Queueing Models MEMORYLESS means that the probability of an event doesn't depend on its past. The above case highlights an example where the past does matter. Examples : Which depend on the past, and which don't? – Throwing dice? – A disk seek distance? – The address of an instruction execution? – The measurement of the length of a table? Prepare to Have Your Mind Bent In An Unexpected Way!! Example: Consider a bus stop where the time between bus arrivals is exponentially distributed with a rate L. Thus the bus arrivals form a Poisson process. If you walk up to the bus stop, how long do you have to wait until the next bus arrives? Queueing Models Example: Consider a bus stop where the time between bus arrivals is exponentially distributed with a rate L. Thus the bus arrivals form a Poisson process. If you walk up to the bus stop, how long do you have to wait until the next bus arrives? 1. Possible solution: Since buses arrive at a rate L, the average time between arrivals is 1/L. But since we walk up at random, we would wait for only half an interval on the average. So we would wait 1/(2L) for the next bus. 2. Possible solution: Since the time between buses is exponentially distributed, it is memoryless. So the residual lifetime for any time that I arrive should be distributed exponentially the same way as the original distribution. Since the average time between buses is 1/L, the average time ( residual ) to wait should also be 1/L. Queueing Models This program generates 1 week's worth of bus arrivals. We assume that the average bus arrival rate is every 12 hours, The arrival rate is 1/12 per hour. We'll take 1 week's worth of arrivals - we will assume that there are 14 arrivals in a week - that's how we set the average arrival rate. Oh - and we assume the buses arrive on the hour (it's simpler that way). // GenerateRandomBusArrivals #include Sorted Random Numbers 2 5 73 81 100 100 102 109 134 136 145 148 152 154 } // Compare routine used in sorting int compare (const void * a, const void * b) { return ( *(int*)a - *(int*)b ); } int main( int argc, char *argv[]) { int Data[MAX_DATA]; int NumberOfSamples = 14; int Index; printf( "This program generates 1 week's worth of bus arrivals.\n"); printf( "We assume that the average bus arrival rate is every 12 hours,\n"); printf( "The arrival rate is 1/12 per hour. We'll take 1 week's worth\n"); printf( "of arrivals - we will assume that there are 14 arrivals in a week \n"); printf( "- that's how we set the average arrival rate.\n"); printf( "Oh - and we assume the buses arrive on the hour (it's simpler that way).\n"); srand((unsigned)time(NULL)); for ( Index = 0; Index < NumberOfSamples; Index++ ) { Data[Index] = rand() % MAX_DATA_VALUE; } qsort (Data, NumberOfSamples, sizeof(int), compare); printf( "\n Sorted Random Numbers\n"); for ( Index = 0; Index < NumberOfSamples; Index++ ) printf( "%d ", Data[Index] ); printf("\n"); Queueing Models So given these actual bus arrivals, how long on average must you wait for a bus? Type Types of Stochastic Processes Event Counting Value Time Between Events Stochastic Random Walk Markov Birth-Death Arbitrary - any new state is possible given the current state. Next state may depend on current & previous states. Future states depend only on current state, not on previous state. Next state depends only on current state. Poisson Memoryless Arrival rates are often Poisson. Arbitrary Arbitrary Arbitrary Memoryless Memoryless Service times are often exponential. ( In other words, they're both random. ) Queueing Models Example Cars on a highway (highly interactive.) Monopoly Game. Monopoly Game. Event driven scheduler. Bus problem above. Customer Arrivals The queue – A place where customers are stored before being serviced. The device doing the actual service of the customers. Customer Departures Queueing Models Arrival process: Service Time Distribution: Number of Servers: System Capacity : Population Size: Service Discipline: The shorthand for queue description is thus The inter-arrival and service times are typically of the following types: M Exponential D Deterministic G General – Memoryless, the distribution we’ve just been talking about. – the times are constant and there is no variance. –distribution is not specified and the results are valid for all distributions Queueing Models If we have a single queue obeying certain properties, we can get all kinds of nice metrics. But, it must have those required properties!! REQUIRED PROPERTIES: • Arrivals are random with a rate of X per time. ( Poisson – when we say this, we mean the inter-arrival time is exponentially distributed. ) [ Note that in steady state, throughput = arrival rate.] Many texts use l for this. • Service times are random with a value of D. (Exponential ) [ Note this is the Demand we've seen before.] Many texts use m for this. The rate of service is m = 1/D. • There's the possibility of an infinite number of customers. • There's a single server. [ So the derivation we’re about to do doesn't work for a multiprocessor CPU.] Queueing Models These are general requirements and hold for many practical applications. Other analysis can be done for cases outside these requirements, but we don't do it here. We will apply a method called local balance to a system like that pictured on the next page: The queue is of type M / M / 1. Queueing Models State with 0 in Queue l l State with 1 in Queue State with 2 in Queue m m For simplification, in this particular case, the utilization U is related to throughput and demand by N 1 i (Remember N = XS) 0 Queueing Models By Definition: serviced. A queue is defined to contain customers that are both waiting and being In an equilibrium state, from the picture below, the following equations can be formed: p i p i m p i = l p i-1 = ( l / m ) p i -1 = ( l / m ) i p 0 = U i p 0 The probability of having i customers in the queue is p i = ( 1 – U ) U i [ Note that p 0 = ( 1 - U ) so p i > 0 = U . But this is just the utilization we defined before.] l l State with 0 in Queue State with 1 in Queue State with 2 in Queue m Queueing Models m The average number of customers in the queue (waiting and being serviced) is N = U ( 1 - U ) From Little's Law ( N = X T ) in steady state, we can derive the average time spent at the queueing center ( both in the queue and being serviced ). Note what happens to this response time as the utilization increases! T = R k = D ( 1 - U ) Queueing Models Example: Pat is designing a communications server that receives requests from "higher level" routines. The requests are collected by a Request Handler that does nothing but put them into buffers. These requests are removed from the buffers by the Request Processor on a first come first serve basis. requests -> Request Handler -> Buffers[n] -> Request Processor -> The requests arrive randomly at a rate of 5/second. The Request Processor can service 10 items per second from the buffer. Since allocating buffers is an expensive business, Pat wants to preallocate an adequate number of buffers so none need be allocated 99% of the time. Clearly there's a tradeoff here between memory usage and time-to-allocate. How many buffers should be preallocated? Queueing Models This is the setup for the case of both of them are filled. M / M / 2. The probability of transition from a lower population to a higher one is the same as before ( arrivals are the same.) But the probability of one of the TWO servers finishing is twice as great when l State with 0 in Queue m State with 1 in Queue l 2m State with 2 in Queue l 2m State with 3 in Queue Queueing Models Utilization: U = X D Probability of n customers in the system: p n = ( 1 – U ) U n Mean number of customers in the system: N = U / ( 1 – U ) Mean time spent in the system: T = D / ( 1 – U ) Probability of finding n or more jobs in the system: U n Mean number of jobs served in one busy period: 1 / ( 1 – U ) Mean busy time duration: D / ( 1 – U ) Queueing Models Goals: • You should be able to create, use, and understand a simple analytical model. • You should have a general idea of when such models are applicable and should understand some of the buzzwords. PERSPECTIVE: An Analytical Model uses mathematical operations to capture the relationships between observable quantities. The computations don't necessarily mimic real actions as they do for simulations. Examples: • The equations we've been using such as Little's Law. • Local Balance Equations - these enumerate the states the system can be in and then determine the transitions between states. (This is what we just did with single queues.) • Mean Value Analysis - an iterative approach using the equations we've already learned. Queueing Models Analytical models can be applied to: • Single Queues • Queueing Networks Queueing Networks are networks of queues; two or more queues tied together. They can be: • Open: - Typical of transaction processing. Jobs enter and leave the system being studied. • Closed : - typical of batch or terminal systems. Jobs always remain somewhere within the system. • Single Class - the customers are indistinguishable from each other; they have similar service demands and routing characteristics. • Multiple Class - several categories of customers can be identified. A batch class, for example, might be heavily CPU bound while a terminal class is I/O bound. To use a multiple class model requires determining the characteristics for EACH of the classes involved. Queueing Models This is EASY!! It's simply an extension of the equations we used for the single queue. Utilization: Throughput: Max. Throughput: U X k k = = X max = X D k X V k 1 D max {D k is the service time) (Throughput = arrivals) Residence Time: R k = D k (delay centers) D k ( 1 - U k ) (queueing centers) Queueing Models Queue Length: System Response Time: Q k = T = R = U k U k ( 1 - U k ) R k (delay centers) (queueing centers) Average Number In System: N = Q = Q k We'll be using the FIGURE on the next page and applying these Laws to the system shown there, at a number of different levels. Remember, N = Number of requests in the "system" X = Throughput R = Residence time per request. S = Service Time Little's law is 90% of all you'll ever need. Queueing Models Terminals CPU DISK C DISK B Queueing Models DISK A Box 1 in the Figure: not including the queue. A single resource, DISK C Here the population, N, is either 1 or 0 ( in use or not ). Utilization is equal to the average number of requests present, or the average N. Throughput X is the number of requests serviced per time. Residence time R is, for this case, the service time. [NOTE: Little's Law reduces to Utilization Law.] Example: Suppose a disk, serves 100 requests/second, with the average request needing 0.008 seconds of disk service. N = X * R U = X * S R = S/(1 – U ) What is the utilization of the disk? Queueing Models Box 2 in the Figure: A single resource, including the queue. Now the population includes both those requests in the queue and in service. Throughput remains the rate that the resource satisfies requests. DISK C Residence time is the queueing and service times. sum of Example: You will need to use the result from the previous slide. What is the time a request spends at the disk subsystem ( spindle + queue)? N = X * R U = X * S R = S/(1 – U ) What is the average number of requests in “Box 2”? For one of these requests, what is the average queueing time, and what is the average service time? Queueing Models Box 3 in the Figure: Central Subsystem, not including terminals. The population now is the number of users with activity in the system - those not "thinking". Throughput is the rate that requests flow between system and users. Residence time is now what we call response time. Example: The average system throughput (entering & leaving Box3) is 0.5/sec. There are an average of 7.5 "ready" (waiting) users in the Box. What is the average response time? CPU DISK C DISK B Queueing Models DISK A Box 4 in the Figure: including terminals. Entire system, Terminals The population is the total number of users, both waiting and thinking. Throughput is the rate that requests flow between system and users. (same as box 3). Residence time is the response time and think time. sum of Example: There are 10 users with average think time of 5 seconds, and the system has average response time of 15 seconds. What is the throughput? Queueing Models For system wide applications of Little's Law, since time represents both thinking and waiting for a response, N SS N ES = X R = X ( R + Z ) for Subsystem or lower. for the Entire system. Example: A System has 64 interactive users, with an average think time of 30 seconds. Two interactions complete on average each second. What is the average response time for this system? How many interactions are in the system at any time? What sized System ( how many CPU's ) would be needed for this application? Queueing Models Example Problems You Should Now Be Able To Do: The average delay experienced by a packet when traversing a computer network is 100 msec. The average number of packets that cross the network is 128 packets/sec. What is the average number of packets in transit in the network? Example Problems You Should Now Be Able To Do: Measurements taken during one hour from a Web server indicate that the utilization of the CPU and the two disks are: U CPU = 0.25, U disk1 = 0.35, and U disk2 = 0.30. The Web server log shows that 21,600 requests were processed during the measurement interval. What are the service demands (the time used by each request) at the CPU and both disks? What is the maximum throughput, and what was the response time of the Web server during the measurement interval? Example Problems You Should Now Be Able To Do: A computer system is measured for 30 minutes. During this time, 5,400 transactions are completed and 18,900 I/O operations are executed on a certain disk that is 40% utilized. What is the average number of I/O operations per transaction on this disk? What is the average service time per transaction on this disk? Solutions on a later page Queueing Models Example Problems You Should Now Be Able To Do: A file server is monitored for 60 minutes, during which time 7,200 requests are completed. The disk utilization is measured to be 30%. The average service time at this disk is 30 msec per IO. What is the average number of accesses to this disk per file request? Example Problems You Should Now Be Able To Do: A computer system has one CPU and two disks: disk 1 and disk 2. The system is monitored for one hour and the utilization of the CPU and of disk 1 are measured to be 32% and 60%, respectively. Each transaction makes 5 I/O requests to disk 1 and 8 to disk 2. The average service time at disk 1 is 30 msec and at disk 2 is 25 msec. Find the system throughput. Find the utilization of disk 2. Find the average service demands at the CPU, disk 1, and disk 2. Example Problems You Should Now Be Able To Do: An interactive system has 50 terminals and the user's think time is equal to 5 seconds. The utilization of one of the system's disk was measured to be 60%. The average service time at the disk is equal to 30 msec. Each user interaction requires, on average, 4 I/Os on this disk. What is the average response time of the interactive system? Queueing Models Solutions on a later page Problem Solution: The average delay experienced by a packet when traversing a computer network is 100 msec. The average number of packets that cross the network is 128 packets/sec. What is the average number of packets in transit in the network? Straight usage of Little’s Law: N = XS = 128 packets/sec * 0.1 sec = 12.8 packets Problem Solution: Measurements taken during one hour from a Web server indicate that the utilization of the CPU and the two disks are: U CPU = 0.25, U disk1 = 0.35, and U disk2 = 0.30. The Web server log shows that 21,600 requests were processed during the measurement interval. What are the service demands (the time used by each request) at the CPU and both disks? What is the maximum throughput, and what was the response time of the Web server during the measurement interval? There are 21,600 requests/hour = 60 requests/sec. During each second, the CPU is used 250 milliseconds – so each of the 60 requests is using 4.16 milliseconds. Similarly the disks use 5.83 milliseconds and 5 milliseconds of service per transaction. Maximum throughput is determined by the device having the highest utilization, that disk is U disk1 = 0.35. When max’d out, there will be 1 / 0.35 more traffic – or 2.86 more. Thus the maximum throughput will be 2.86 * 60 transactions/second = 171 transactions/second. You can determine the total response time by calculating the response time at each queueing center. This is 4.16 msec/(1 – 0.35) + 5.83 msec / ( 1 – 0.35) + 5.0 msec / ( 1- 0.3) = 6.4 + 8.97 + 7.1 = 22.5 msec Queueing Models Problem Solution: A computer system is measured for 30 minutes. During this time, 5,400 transactions are completed and 18,900 I/O operations are executed on a certain disk that is 40% utilized. What is the average number of I/O operations per transaction on this disk? What is the average service time per transaction on this disk? Put everything into the same time units – (seconds usually work best) 5,400 transactions / 30 minutes = 3 transactions/second 18,900 IO / 30 minutes = 10.5 IOs / second So the number of IOs/transaction = 10.5 / 3 = 3.5 IOs / transaction In each second, this disk is busy 400 milliseoncds of time. During a second, 3 transactions complete. So the disk service time per transaction is 133 milliseconds. Problem Solution: A file server is monitored for 60 minutes, during which time 7,200 requests are completed. The disk utilization is measured to be 30%. The average service time at this disk is 30 msec per IO. What is the average number of accesses to this disk per file request? The idea is that a file server request may result in multiple IOs to the disk necessarily. 7,200 requests / 3,600 seconds = 2 requests/second. The disk is 30% busy so it runs for 300 milliseconds each second. – it’s not a 1 to 1 match The service time is 30 milliseconds, so 10 IOs are completed each second. 10 IOs for 2 requests means there are on average 5 IOs / request. Queueing Models Problem Solution: A computer system has one CPU and two disks: disk 1 and disk 2. The system is monitored for one hour and the utilization of the CPU and of disk 1 are measured to be 32% and 60%, respectively. Each transaction makes 5 I/O requests to disk 1 and 8 to disk 2. The average service time at disk 1 is 30 msec and at disk 2 is 25 msec. Find the system throughput. Find the utilization of disk 2. Find the average service demands at the CPU, disk 1, and disk 2. It turns out here that disk 1 is where we have the most information: For this disk, it’s busy 600 milliseconds out of each second; it takes 30 milliseconds for each IO; so there are 20 IOs/second. Since each transaction makes 5 IO requests to disk 1, that means there are 4 transactions/second. Once we have this answer, we know there are 32 IOs to disk 2 per second. Each of those IOs takes 25 milliseconds, giving a total time usage of 800 milliseconds. That means this disk is 80% utilized. Problem Solution: An interactive system has 50 terminals and the user's think time is equal to 5 seconds. The utilization of one of the system's disk was measured to be 60%. The average service time at the disk is equal to 30 msec. Each user interaction requires, on average, 4 I/Os on this disk. What is the average response time of the interactive system? The disk is doing 20 IOs / second. Each transaction is 4 IOs, so the throughput is 5 trans/sec. To get this throughput requires each of the 50 users is executing a transaction every 10 seconds. Thus the time in the system must be 5 seconds (because the user is already thinking for 5 secs.) Queueing Models FORCED FLOW LAW: The Forced Flow Law states that the flow in all parts of a system must be consistent. Suppose we count both system completions and also completions at each resource. The visit count ( visit ratio ) is defined as: Resource Completion: System Completion: Visit Count: C k C V k = C k / C k k Queueing Models Suppose a system has: 30 terminals ( N = 30 ). 18 seconds average think time ( Z = 18 ). 20 visits to a specific disk/interaction (V disk 30% utilization of that disk (U disk = 0.30 ). = 20 ). 25 millisecs is the average service required per visit to the disk (S disk 0.025 sec.). = We want to know : Disk throughput System throughput Response time = = = RESPONSE TIME LAW: R = N/X - Z Queueing Models THE FORCED FLOW LAW: X k = V k X Consider the problem of a spy from Burger King trying to figure out how many people are at a McDonald's. The spy can't sit inside and watch all day, so must somehow calculate the number from information obtained from outside observations. Thirty customers/hour arrive on the average ( over a long period of time ) and the average customer exits after 12 minutes. Assuming that all this time is spent standing in line, what is the mean queue length in the restaurant? If the Standard Deviation of the 30 customers is 3, what is the uncertainty of the queue length? What happens if both the arrival rate and the service time have uncertainties? RESPONSE TIME LAW: R = N/X - Z Queueing Models THE FORCED FLOW LAW: X k = V k X Model Inputs: V cpu S cpu D cpu = 121 V disk1 = 70 = 0.005 S disk1 = 0.030 = 0.605 c D disk1 = 0.3 jobs/sec = 2.1 V disk2 = 50 S disk2 = 0.027 D disk2 = 1.35 Model Structure: Disk1 Departures Arrivals Disk2 CPU Queueing Models Model Inputs: V cpu S cpu D cpu = 121 V disk1 = 0.005 S disk1 = 70 V = 0.030 S disk2 disk2 = 0.605 D disk1 c = 2.1 D = 0.3 jobs/sec disk2 = 50 = 0.027 = 1.35 Model Structure: Arrivals Departures Disk1 Disk2 CPU Model Outputs: l = 1 / D max = 1 / 2.1 = 0.476 jobs/sec X cpu (0.3) = c V cpu U cpu (0.3) = c D cpu = (0.3)(121) = (0.3)(0.605) R cpu (0.3) = D cpu / ( 1 – U cpu (0.3) ) Q cpu (0.3) = U cpu (0.3) / ( 1 – U cpu (0.3) ) = 36.3 visits/sec = 0.182 = 0.605 / 0.818 = 0.740 secs = 0.182 / 0.818 = 0.222 jobs R(0.3) = R cpu (0.3) + R disk1 (0.3) + R disk2 (0.3) = 0.740 + 5.676 + 2.269 = 8.685 secs Q(0.3) = l R( l ) = (0.3) (8.685) = 2.606 jobs Queueing Models T A C W B Z V k is the length of TIME we observed the system. is the number of request ARRIVALS observed. is the number of request DEPARTURES observed. is the ACCUMULATED TIME for all requests within the system – time spent both waiting for and using resources. is the length of time that the resource was observed to be BUSY. is the think time of a terminal user. the visit ratio, is the number of times device k is visited per transaction. Arrival Rate Throughput (Departure Rate) Utilization Service Requirement Requests in system Residence time Y = A / T X = C / T U = B / T S = B / C N = W / T R = W / C UTILIZATION LAW U = X S LITTLE'S LAW N = X R RESPONSE TIME LAW R = N/X – Z THE FORCED FLOW LAW: X k = V k X Queueing Models Goals: To increase facility using performance metrics. To be able to calculate limits or boundaries on performance metrics. To be able to do "back of the envelope" calculations. UTILIZATION LAW U = X S LITTLE'S LAW N = X R RESPONSE TIME LAW R = N/X – Z THE FORCED FLOW LAW: X k = V k X Queueing Models BOUNDARY VALUES: The goal here is to make estimations of the outside limits of a performance parameter. We do this by selective "blind" application of our simple laws to complex systems; our results give upper and lower bounds, not an exact answer. • • • Example: An editor program wants to read 100 disk pages into memory. required for each disk read are: The resources 5 milliseconds of CPU Processor time. 4 milliseconds of Controller Processor time. 2 milliseconds of SCSI Handshaking time. 10 milliseconds of DISK seek/rotation/transfer time. What is the bottlenecking device? Warning!! This is a trick question! What is the worst throughput for this system, assuming the editor single threads the reads - waiting until each read completes before starting the next? What is the best throughput for this system? How long will it take to read in the 100 disk blocks? Queueing Models BOUNDARY VALUES: Now that we know the limits for one user, let's expand this for N users . • • • • Example: Multiple users, each with their own copy of the editor, are reading files from the same disk (parameters are the same as the last example.) What is the bottleneck device in this case? Considering only the bottleneck device, what is the best throughput achievable? Now taking into account all devices, what is the BEST throughput N users can achieve ( assuming the requests NEVER get in each other's way?) Now taking into account all devices, what is the WORST throughput N users can achieve ( assuming the requests ALWAYS get in each other's way?) Queueing Models BOUNDARY VALUES: We define DEMAND as the amount of a resource used for a transaction; this is just the number of visits to that resource during one transaction, and the service time used at each of those visits: D D k D total max = V k * S k = D = S D k = D bottleneck SUMMARY: Best throughput overall: Best throughput for N users: X = 1 / D Bottleneck X = N / D Worst throughput for N users: X = N / ( N * D ) = 1 / D Queueing Models Exercise: Consider an interactive system with a CPU and two disks. The following measurement data was obtained system: by measuring the Determine the visit counts V k , service times per visit S k , and service demands D k at each center. ----------- Observation Data --------------- Observation interval Active terminals 30 minutes 30 Think time 12 seconds Completed transactions 1,600 Fast disk accesses 32,000 Slow disk accesses CPU busy Fast disk busy Slow disk busy Data At Centers 12,000 1,080 seconds 400 seconds 600 seconds V k S k D k CPU Fast Disk Slow Disk D = S D k - - Queueing Models Exercise: Give optimistic and pessimistic asymptotic throughput and response time for 10 and 40 active terminals. Optimistic X Pessimistic X Optimistic R Pessimistic R bounds on ----------- Observation Data --------------- Observation interval Active terminals 30 minutes 30 Think time 12 seconds Completed transactions 1,600 Fast disk accesses 32,000 Slow disk accesses CPU busy Fast disk busy Slow disk busy 12,000 1,080 seconds 400 seconds 600 seconds Bounds on Throughput and Response Time Metric -- 10 - Terminal Count -- 40 - 1 / D max N / {D+Z} N / { N * D + Z } D N * D max - Z N * D Queueing Models Exercise: Consider the following changes to the system: • Move all files to the fast disk. • Replace the slow disk by a second fast disk. • Increase the CPU speed by 50% (with the original disks.) ----------- Observation Data --------------- Observation interval Active terminals 30 minutes 30 Think time 12 seconds Completed transactions 1,600 Fast disk accesses 32,000 Slow disk accesses CPU busy Fast disk busy Slow disk busy 12,000 1,080 seconds 400 seconds 600 seconds • Increase the CPU speed by 50% and balance the disk load across 2 fast disks. What changes should be made so response time will not exceed 10 seconds? Rank the changes. efficacy of these Queueing Models This is an extension of the Operational Analysis we did before. This is the method that should be used on closed systems. Assumptions: • Applies to various service disciplines and service time distributions. (We will only do fixed capacity centers here.) • The arrival rate is not dependent on the load. • Arrivals are a Poisson process. We will use a technique called Mean Value Analysis that is based on the repetitive application of three equations: 1. Little's Law applied to the whole network: 2. Little's Law applied to each service center: 3. The service center residence time equations: Menasce, et. al. do this in Section 12.3 of the text. A great book on this, now out of print is at http://www.cs.washington.edu/homes/lazowska/qsp/ Queueing Models 1. Little's Law applied to the whole network: X ( N ) = N Z k K = 1 R k ( N ) where X(N) is the system throughput, Z there are N customers in the network. is the think time, and R k (N) the residence time at center k when 2. Little's Law applied to each service center: Q k ( N ) = X ( N ) R k ( N ) 3. The service center residence time equations: R k ( N ) = D k R k ( N ) = D k 1 A k ( N ) where A k (N) Delay Centers Queueing Centers is the average number of customers seen at center k when a new customer arrives. The number seen on arrival (in equilibrium) is equal to the average queue length when there is one less customer in the system! So the equations bootstrap!!! A k ( N ) = Q k ( N 1 ) Queueing Models //////////////////////////////////////////////////////////////////////////////// // This program does very simple mean value analysis. //////////////////////////////////////////////////////////////////////////////// #define MAX_NUMBER_OF_CENTERS 30 int number_of_customers; int number_of_centers; /* # of centers requested by the user. */ int number_of_customers; /* The population in each class */ float input; double think_time; /* The time spent at the terminal */ double demand[MAX_NUMBER_OF_CENTERS];/* Service demand of a center. */ double q_length[MAX_NUMBER_OF_CENTERS]; double residence_time[MAX_NUMBER_OF_CENTERS+1]; // Residence time at a center. double throughput; double system_response_time; // Prototypes void print_the_numbers(int n); This program is at: http://web.cs.wpi.edu/~jb/perf/Lectures/mva.c Queueing Models main() { char input_string[80]; int center, n; //////////////////////////////////////////////////////////////////////////// // Get the input information from the user. //////////////////////////////////////////////////////////////////////////// printf( "Number of users: "); gets( input_string ); number_of_customers = atoi( input_string ); printf( "Number of centers: "); gets( input_string ); number_of_centers = atoi( input_string ); printf( "Think Time: "); gets( input_string ); sscanf( input_string, "%f", &input ); think_time = (double)input; for ( center = 0; center < number_of_centers; center++ ) { printf( " Demand for center %d: ", center); gets( input_string ); sscanf( input_string, "%f", &input ); demand[center] = (double)input; } //////////////////////////////////////////////////////////////////////////// // Now that the network is described, we perform the evaluation. // Begin by initializing to the trivial solution for 0 customers. //////////////////////////////////////////////////////////////////////////// for ( center = 0; center < number_of_centers; center++ ) { q_length[center] = 0; } Queueing Models //////////////////////////////////////////////////////////////////////////// // The algorithm now iterates through each population value n. //////////////////////////////////////////////////////////////////////////// for ( n = 0; n <= number_of_customers; n++ ) { system_response_time = 0; for ( center = 0; center < number_of_centers; center++ ) { residence_time[center] = demand[center] * ( 1.0 + q_length[center] ); system_response_time += residence_time[center]; } //////////////////////////////////////////////////////////////////////////// // Next use Little's Law to compute the system throughput. //////////////////////////////////////////////////////////////////////////// throughput = n / ( think_time + system_response_time ); //////////////////////////////////////////////////////////////////////////// // Finally use Little's Law to compute center queue lengths. //////////////////////////////////////////////////////////////////////////// for ( center = 0; center < number_of_centers; center++ ) { q_length[center] = residence_time[center] * throughput; } print_the_numbers(n); } /* END OF LOOP THROUGH POPULATION */ exit(0); } /* END OF MAIN */ Queueing Models //////////////////////////////////////////////////////////////////////////////// // Print the results of each loop //////////////////////////////////////////////////////////////////////////////// void print_the_numbers(int n) { int i; printf( "\n" ); printf( "Iteration: %d System Response Time: %f Throughput: %f\n", n, system_response_time, throughput ); for ( i = 0; i < number_of_centers; i++ ) printf( "Center: %d Residence Time: %f Queue Length: %f\n", i, residence_time[i], q_length[i] ); } // END OF PRINT_THE_NUMBERS Queueing Models Model Inputs: Model Structure: V cpu S cpu D cpu = 121 V disk1 = 70 = 0.005 S disk1 = 0.030 = 0.605 c D disk1 = 0.3 jobs/sec = 2.1 V disk2 = 50 S disk2 = 0.027 D disk2 = 1.35 Disk1 Terminals Disk2 CPU The INPUT numbers are the same as were used for the open model. We modify the picture so there are no “arrivals” or “departures”, but instead there are terminals – the jobs stay in the picture (which is what makes it closed.) N = 3, Z = 15 . Model Inputs: V cpu S cpu D cpu = 121 V disk1 = 0.005 S disk1 = 70 V = 0.030 S disk2 disk2 = 0.605 D disk1 c = 2.1 D = 0.3 jobs/sec disk2 = 50 = 0.027 = 1.35 X ( N ) = N Z k K = 1 R k ( N ) Q k ( N ) = X ( N ) R k ( N ) To start with, we initialize the queue lengths to 0: R k ( N ) = D k R k ( N ) = D k 1 A k ( N ) Q cpu = Q disk1 = Q disk2 = 0 A k ( N ) = Q k ( N 1 ) Iteration Number 1: R cpu = 0.605 sec R disk1 X = 1 / ( 15 + 4.055 ) = 2.1 sec = 0.0525 Q cpu = 0.0318 Q disk1 = 0.1102 R disk2 = 1.35 sec. Q disk2 = 0.0708 Iteration Number 2: R cpu = 0.624 sec X = 0.1030 Q cpu = 0.0643 R disk1 = 2.331 sec R disk2 = 1.446 sec. Q disk1 = 0.2403 Q disk2 = 0.1490 Queueing Models Model Inputs: V cpu S cpu D cpu = 121 V disk1 = 0.005 S disk1 = 70 V = 0.030 S disk2 disk2 = 0.605 D disk1 c = 2.1 D = 0.3 jobs/sec disk2 = 50 = 0.027 = 1.35 X ( N ) = N Z k K = 1 R k ( N ) Q k ( N ) = X ( N ) R k ( N ) Iteration Number 3: R k ( N ) = D k R k ( N ) = D k 1 A k ( N ) R cpu = 0.644 sec X = 0.1515 Q cpu = 0.0976 R disk1 = 2.605 sec Q disk1 = 0.3947 R disk2 = 1.551 sec. Q disk2 = 0.2350 A k ( N ) = Q k ( N 1 ) From these numbers we can calculate: X(3) R(3) Q(3) = ( N / X ) - Z = N - X(3) Z = 0.1515 = 4.74 = 0.72 There’s a very nice example on Jain, pp 578-9. Queueing Models This section has discussed: Queueing Lingo all the definitions you’ll ever need! Queueing Analytical Models especially the Single Queue in DETAIL Operational Laws how to solve a number of complex problems using the equations we already know and love. Works especially well for open models. drawing conclusions about limits - what to do when we can’t solve the math exactly. Mean Value Analysis Using the equations in an iterative fashion to solve closed models. Queueing Models11
Lab - You Get To See It Happen Before Your Very Eyes!
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Queueing Lingo
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Queueing Lingo
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Queueing Lingo
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Queueing Lingo
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Queueing Lingo
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PROPERTIES OF QUEUES
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PROPERTIES OF QUEUES How do we describe a queue? These are the important aspects:
A / S / m / B / K / SD.
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THE SINGLE QUEUE
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THE SINGLE QUEUE
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THE SINGLE QUEUE X =
X =
= 1/D
= 1/D
Note:
p
= U , U = X D
U
= l / m
p
= ( 1 – U )
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THE SINGLE QUEUE
X =
X =
= 1/D
= 1/D 23
THE SINGLE QUEUE
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THE SINGLE QUEUE
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THE SINGLE QUEUE
X =
= 1/D
X =
X =
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THE SINGLE QUEUE The following equations hold for the single queue.
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ANALYTICAL MODEL
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ANALYTICAL MODEL
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ANALYTICAL MODEL SINGLE CLASS OPEN QUEUEING NETWORK MODEL SOLUTIONS:
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ANALYTICAL MODEL SINGLE CLASS OPEN QUEUEING NETWORK MODEL SOLUTIONS:
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4 ANALYTICAL MODEL
1 3
2
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ANALYTICAL MODEL
1
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ANALYTICAL MODEL
2
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ANALYTICAL MODEL
3
2
35
4 3 ANALYTICAL MODEL
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ANALYTICAL MODEL
RESPONSE TIME LAW: R = N/X - Z
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ANALYTICAL MODEL
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ANALYTICAL MODEL
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ANALYTICAL MODEL
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ANALYTICAL MODEL
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ANALYTICAL MODEL
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ANALYTICAL MODEL
THE FORCED FLOW LAW: X
= V
X
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ANALYTICAL MODEL Example:
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ANALYTICAL MODEL Example:
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ANALYTICAL MODEL EXAMPLE OF THE SOLUTION OF AN OPEN MODEL:
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Solution Of An Open Model EXAMPLE:
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SUMMARY OF
PERFORMANCE METRICS
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OPERATIONAL LAWS
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OPERATIONAL LAWS
50
OPERATIONAL LAWS
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OPERATIONAL LAWS
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OPERATIONAL LAWS
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OPERATIONAL LAWS
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OPERATIONAL LAWS
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MEAN VALUE ANALYSIS
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MEAN VALUE ANALYSIS
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MEAN VALUE ANALYSIS
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MEAN VALUE ANALYSIS
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MEAN VALUE ANALYSIS
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MEAN VALUE ANALYSIS
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MEAN VALUE ANALYSIS EXAMPLE OF THE SOLUTION OF A CLOSED MODEL:
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EXAMPLE: Solution Of An Open Model
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EXAMPLE: Solution Of An Open Model
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CONCLUSION
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