29:null 2a:["$","title",null,{"children":"Performance Analysis With PDQ | Amédée d'Aboville | Observable"}] 2f:[] 2d:{"id":"ac74751d44174190","slug":"chapter-4-getting-the-jump-on-queuing","title":"Performance Analysis With PDQ","thumbnail":"9854a04f6de2b26cece79f48edc344e84eb6ce1eddd0150feae0a5c722f02edd","access_level":"public","create_time":"$D2021-01-15T23:18:47.271Z","update_time":"$D2021-01-20T23:53:42.275Z","view_time":null,"first_public_time":"$D2021-01-17T23:47:32.127Z","has_unpublished_edits":null,"fork":false,"stars":0,"starred":false,"owner":{"type":"team","id":"fcbc8e919fc88c92","login":"amedeedaboville","name":"Amédée d'Aboville","avatar_url":"https://avatars-next.observableusercontent.com/c0af1c83f1b62e8774770a7843123048002196239dca1f55575fd3661685cad5","tier":"starter_2024"},"version":521,"publish_version":521,"role":null,"authored":false,"authors":[{"type":"individual","id":"eed87986fc42c86f","login":"amedeedaboville","name":"Amédée d'Aboville","avatar_url":"https://avatars-next.observableusercontent.com/c0af1c83f1b62e8774770a7843123048002196239dca1f55575fd3661685cad5","tier":null}],"members":[{"type":"individual","id":"eed87986fc42c86f","login":"amedeedaboville","name":"Amédée d'Aboville","avatar_url":"https://avatars-next.observableusercontent.com/c0af1c83f1b62e8774770a7843123048002196239dca1f55575fd3661685cad5","tier":null,"role":"editor"}],"comments":[],"custom_thumbnail":false,"trashed":false,"lapsed":null,"subscribed":null,"forks":0,"collection":null,"type":"notebook","body":{"title":"Performance Analysis With PDQ","cells":[{"id":490,"value":"md`# Performance Analysis With PDQ\nThis is a WIP of every listing the book translated from Perl to Javascript.\n\n- [x] - Listing 2.1 is in [this notebook](https://observablehq.com/d/8c5c68ea2d029b80)\n- ~~Chapter 3 Listings are not done~~\n- [x] Chapter 4\n- [x] Chapter 5\n\nI am translating PDQ to javascript and will also add the PDQ models here\n`","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":0,"value":"md`# Chapter 4 - Getting the Jump on Queuing\n\nThis observable notebook is editable: you can tweak the input variables (busyData, time periods) and all the\nvalues will be recomputed\n`","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":116,"value":" d3 = require('d3')","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":4,"value":"md`### Listing 4.1`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":13,"value":"busyData = [1.23, 2.01, 3.11, 1.02, 1.54, 2.69, 3.41, 2.87, 2.22, 2.83];\n","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":15,"value":"timePeriod = 30 // minutes","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":17,"value":"arrivalCount = busyData.length;","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":7,"value":"arrivalRate = arrivalCount / timePeriod;","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":25,"value":"md`### Listing 4.2 - Calculate the system throughput`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":47,"value":"completionCount = busyData.length","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":49,"value":"throughput = completionCount/ timePeriod","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":56,"value":"normalizedThroughput = `1 Customer every ${1/throughput} minutes`","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":58,"value":"md`### Listing 4.3 - Calculate the service time`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":86,"value":"md`Aggregate busy time (minutes):`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":75,"value":"busyServer = busyData.reduce((a,b) => a+b) ","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":66,"value":"completionsServer = busyData.length","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":68,"value":"serviceTime = busyServer / completionsServer // in minutes","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":99,"value":"md`### Listing 4.4 - Calculate the utilization`","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":106,"value":"utilization = busyServer / timePeriod","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":113,"value":"md`The utilization is ${d3.format('0.2%')(utilization)}`","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":128,"value":"md`### Listing 4.5 - Calculate the delays`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":129,"value":"residenceTime = serviceTime / (1 - utilization) //minutes","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":132,"value":"queueLength = utilization / (1 - utilization)","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":134,"value":"waitingTime = queueLength * serviceTime // minutes","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":138,"value":"md`### Listing 4.6 - Calculate the Erlang B and C functions`","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":145,"value":"servers = 8","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":147,"value":"erlangs = 4","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":153,"value":"rho = erlangs / servers","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":149,"value":"erlangB = {\n let erlangB = erlangs / (1 + erlangs);\n for (let m = 2; m <= servers; m++) {\n erlangB = erlangB * erlangs / (m + (erlangB * erlangs))\n }\n return erlangB\n}","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":170,"value":"erlangC = erlangB / ( 1 - rho + (rho * erlangB))","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":171,"value":"normalizedWaitingTime = erlangC / (servers * (1 - rho))","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":179,"value":"normalizedResponseTime = 1 + normalizedWaitingTime //Exact solution","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":184,"value":"md`The following is the approximation to the M/M/m queue from Equation 4.68:`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":180,"value":"approximateNormalizedResponseTime = 1 / (1 - Math.pow(rho, servers))","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":193,"value":"md`### Listing 4.7 - Repairman model\nThe values are set to the example 4.14, a timeshare compiling system with 230 users, a 300s think time, and mean compile time of 0.63 compiles/second. Try changing them to the example in 4.8.3, by setting n = 430, and then also setting m = 2 and S = 0.66.\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":194,"value":"N = 230","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":197,"value":"Z = 300","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":198,"value":"m = 1","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":200,"value":"S = 0.63","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":202,"value":"L = {\n let p0 = 1;\n let p = 1;\n let L = 0;\n for (let k = 1; k <= N; k++) {\n p *= (N - k + 1) * S / Z;\n p /= Math.min(k, m);\n p0 += p;\n if (k > m) {\n L += p * (k - m);\n }\n }\n L *= 1/p0;\n return L\n}","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":204,"value":"W = L * ( S + Z) / (N - L) // Waiting time","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":206,"value":"R = W + S // Residence time","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":217,"value":"X = N / (R + Z) //throughput","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":253,"value":"U = X * S // Number in service (Little's law)","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":260,"value":"rho_ = U / m // Utilization (from 0 to 1)","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":290,"value":"md`### Listing 5.1 - Passport Renewal Model`","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":292,"value":"appsPerHour = 15;","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":294,"value":"lambda = appsPerHour / 3600","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":296,"value":"passportResults = {\n let p12 = 0.3;\n let p13 = 0.7;\n let p23 = 0.2;\n let p32 = 0.1;\n let L2 = lambda * (p12 + (p32*p13)) / ( 1 - (p32 * p23));\n let L3 = p13 * lambda + p23 * L2;\n \n let rho1 = lambda * 20;\n let rho2 = L2 * 600;\n let rho3 = L3 * 300;\n let rho4 = lambda * 60;\n let Q1 = rho1 / ( 1 - rho1);\n let Q2 = rho2 / ( 1 - rho2);\n let Q3 = rho3 / ( 1 - rho3);\n let Q4 = rho4 / ( 1 - rho4);\n \n let R = (Q1 + Q2 + Q3 + Q4)/ lambda;\n return [ L2/ lambda, L3/ lambda, rho1, rho2, rho3, rho4]\n}","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":298,"value":"md`### Listing 5.2 - MVA Algorithm`","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":306,"value":"mva = (N, D, Z) => { // D is service demand at each queue\n const K = D.length;\n let Q = Array(K).fill(0);\n let R = Array(K).fill(0);\n \n for (let n = 1; n <= N; n++) {\n //1. Calculate the residence time at k\n for (let k = 0; k < K; k++) {\n R[k] = D[k] * (Q[k] + 1);\n }\n //2. Calculate system response time\n let rtt = Z;\n for (let k=0; k {\n const K = D.length;\n const threshold = 0.001;\n let Q = Array(K).fill(N / K);\n let Qprev = Array(K).fill();\n let R = Array(K).fill();\n\n \n do {\n for (let k = 0; k < K; k++) {\n Qprev[k] = Q[k];\n R[k] = D[k] * (1 + Q[k] * (N - 1) / N);\n }\n let rtt = Z;\n for (let k = 0; k < K; k++) {\n rtt += R[k]; \n }\n \n const X = N / rtt;\n for(let k = 0; k < K; k++) {\n Q[k] = X * R[k]; \n }\n } while(Q.some((q, idx) => (Math.abs(q - Qprev[idx]) / q) > threshold))\n return [Q, R, X]\n}","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":376,"value":"approxMva(230, [0.63], 300) //these are very close to the exact solution","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":397,"value":"approxMva(1, [0.63], 8) //these are very close to the exact solution","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":382,"value":"import { chart} from '@joshdata/simple-line-chart'","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":450,"value":"md`Note, the throughput numbers never change, I'm not sure why. I copied the book's implementation.`","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":384,"value":"chart({\n xAxis: {\n title: \"Users(N)\",\n min: 1,\n max: 50\n },\n yAxis: {\n title: \"Response Time R(N)\",\n min: 0,\n max: 18,\n },\n series: [\n { title: \"Exact\", points: n => mva(n, [0.5] ,8)[1][0] },\n { title: \"approximate\", points: n => approxMva(n, [0.5] ,8)[1][0] }\n ]\n})","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":485,"value":"md`### Listing 5.4 - Time-share queuing model`","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":486,"value":"{\n const m = 2;\n const N = 4;\n const D = 5;\n const Z = 0.001;\n let p = 1;\n let p0 =1;\n let L = 0;\n \n for (let k=1; k <=N; k++) {\n p *= ( N -k + 1) * D / Z;\n p /= Math.min(k, m);\n p0 += p;\n if (k > m) {\n L += p * (k - m); \n }\n }\n L *= 1/p0;\n return [m, N, ,D, Z, L])\n}ie","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"}],"files":"$W2f","stdlib":"1","titling":"auto"}} 2e:"$2d:body"