InstructionsChapter 16Operations Research Models and MethodsPaul A. Jensen and Jonathan F. BardQueuing Modelshttp://www.me.utexas.edu/~jensen/ORMM/computation/unit/stoch_mod/index.htmlhttp://www.me.utexas.edu/~jensen/ORMM/computation/unit/stoch_anal/index.htmlhttp://www.me.utexas.edu/~jensen/ORMM/computation/unit/que_add/index.htmlhttp://www.me.utexas.edu/~jensen/ORMM/computation/unit/rvadd/index.htmlInstructionsTo use this workbook, you must first load the Queuing and the Random Variables add-ins. See the Excel help file topic "About add-ins" before continuing.The workbook also uses the Stochastic Analysis and Random Variables add-ins. Load these add-ins.The Queuing add-in and the Random Variables add-in provide functions for analysis. Choose the Relink command from the Queues menu and the Relink Functions command from the Random Variables add-in.This workbook comes without control buttons. To make the buttons choose the Add Buttons commands from the Stochastic Models and the Stochastic Analysis add-ins.Sect.Queuing ModelsLink to Worksheet1System CharacteristicsAdd-ins UsedCharacteristicsComputing System Characteristics from State ProbabilitiesQueues2Markov Queuing SystemsInfinite QueuesQueuing Systems allowing Infinite QueuesQueuesFinite QueuesQueuing Systems with Finite QueuesQueues/Random VariablesFinite PopulationsQueuing Systems with Finite PopulationsQueuesRepair ModelBirth Death Model of Finite Population Repair ShopStochastic ModelsRepair MatrixMarkov Process Rate Matrix for the Repair ProblemStochastic AnalysisRepair EconomicsEconomic Matrix for the Repair Problem (all zeroes)Stochastic AnalysisSteady StateSteady-State Probabilities for the Repair ProblemStochastic AnalysisRepair Model (2)Birth Death Model of Finite Population Repair Shop with Two ServersStochastic ModelsRepair Matrix (2)Markov Process Rate Matrix for the Repair Problem with Two ServersStochastic AnalysisRepair Economics (2)Economic Matrix for the Repair Problem with Two Servers (all zeroes)Stochastic AnalysisSteady State (2)Steady-State Probabilities for the Repair Problem with Two ServersStochastic Analysis3Non-Markov SystemsNon-MarkovValue of Reduction in Service and Arrival Process VariabilityQueues

&APage &PCharacteristicsInfinite QueuesFinite QueuesFinite PopulationsRepair ModelRepair MatrixRepair MatrixRepair EconomicsSteady StateRepair Model (2)Repair Matrix (2)Repair Matrix (2)Repair Economics (2)Steady State (2)Non-Markovhttp://www.me.utexas.edu/~jensen/ORMM/computation/unit/stoch_mod/index.htmlhttp://www.me.utexas.edu/~jensen/ORMM/computation/unit/stoch_anal/index.htmlhttp://www.me.utexas.edu/~jensen/ORMM/computation/unit/que_add/index.htmlhttp://www.me.utexas.edu/~jensen/ORMM/computation/unit/rvadd/index.html

CharacteristicsComputing System Characteristics from State ProbabilitiesTable 1State, n012345.00006Probability, pn0.0680.170.2120.1770.1470.12300.1031Queue StationQue1Arrival Rate5Service Rate/Channel2. What is the probability that all servers are idle?Number of Servers3Pr{all servers idle} = p0 = 0.0680.0680Max. Number in System62. What is the probability that a customer will not have to wait?Number in Population***Pr{no wait} = p0 + p1 + p2 = 0.4500.4500TypeM/M/3/63. What is the probability that a customer will have to wait in the queue?Mean Number at Station2.9444885254Pr{wait} = 1 P(no wait) = 0.5500.5500Mean Time at Station0.65609252454. What is the probability that an arriving customer is lost?Mean Number in Queue0.7005301504Pr{lost customer} = p6 = 0.1020.1030Mean Time in Queue0.15609251095. What is the expected number of customers in the queue?Mean Number in Service2.2439582348Lq = 1p4 + 2p5 + 3p6 = 0.7000.7020Mean Time in Service0.56. What is the expected number in service?Throughput Rate4.4879164696Ls = p1 + 2p2 + 3(1 p0 p1 p2) = 2.2442.2440Efficiency0.74798607837. What is the expected number in the system?Probability All Servers Idle0.0679588094L = Lq + Ls = 2.9442.9460Prob. All Servers Busy0.54977277428. What is the efficiency (utilization) of the servers?Prob. System Full0.1024166793E = Ls/s = 2.244/3 = 74.8%.0.7480P(0)0.0679588094P(1)0.1698970236Example 1P(2)0.2123712795Lambda =5P(3)0.1769760662Lambda_Bar=4.485P(4)0.1474800517Ws0.5003344482P(5)0.1229000401Wq =0.1565217391P(6)0.1024166977W=0.6568561873P(7)0P(8)0Example 2P(9)0AnalysisP(10)0l = 2 customers/min (given)2W = 3 min (given)3Ws = 1.25 min (given)1.25m = 0.8 customers/min0.8L = lW = 6 customers6Wq = W Ws = 1.75 min1.75Lq = lWq = 3.5 customers3.5Ls = lWs = 2.5 customers2.5E = Ls/s = 2.5/3 83.3%0.8333333333

Infinite QueuesQueuing Systems allowing Infinite QueuesTable 2Queue StationWorkStationQueue StationWorkstation2Queue StationTelephone_1Telephone_2Telephone_3Arrival Rate1.5Arrival Rate1.5Arrival Rate101010Service Rate/Channel2Service Rate/Channel2Service Rate/Channel111Number of Servers1Number of Servers2Number of Servers111213Max. Number in System***Max. Number in System***Max. Number in System*********Number in Population***Number in Population***Number in Population*********TypeM/M/1TypeM/M/2TypeM/M/11M/M/12M/M/13Mean Number at Station3Mean Number at Station0.8727272749Mean Number in Queue6.82118596872.24693954670.9509019036Mean Time at Station2Mean Time at Station0.5818181634Mean Time in Queue0.68211859690.22469395470.0950901904Mean Number in Queue2.25Mean Number in Queue0.1227272701Efficiency0.90909093620.83333331350.769230783Mean Time in Queue1.5Mean Time in Queue0.0818181801Critical Wait Time111Mean Number in Service0.75Mean Number in Service0.75P(Wait >= Critical Wait)0.25093739450.06081803340.0142027874Mean Time in Service0.5Mean Time in Service0.5Throughput Rate1.5Throughput Rate1.5Efficiency0.75Efficiency0.375Probability All Servers Idle0.25Probability All Servers Idle0.4545454383Prob. All Servers Busy0.75Prob. All Servers Busy0.2045454532Prob. System Full0Prob. System Full0CumCritical Wait Time1Critical Wait Time11P(Wait >= Critical Wait)0.4548979948CumP(Wait >= Critical Wait)0.0167901132CumP(0)0.250.25P(0)0.45454543830.4545454383P(1)0.18750.4375P(1)0.34090907870.795454517P(2)0.1406250.578125P(2)0.12784090450.92329542150.0767045785P(3)0.105468750.68359375P(3)0.04794033920.97123576070.0287642393P(4)0.07910156250.76269531250.2373046875P(4)0.01797762720.98921338790.0107866121P(5)0.05932617190.82202148440.1779785156P(5)0.00674161020.9959549981P(6)0.04449462890.8665161133P(6)0.00252810380.9984831019P(7)0.03337097170.899887085P(7)0.00094803890.9994311409P(8)0.02502822880.9249153137P(8)0.00035551460.9997866555P(9)0.01877117160.9436864853P(9)0.0001333180.9999199735P(10)0.0140783787P(10)0.00004999420.9999699677

Finite QueuesQueuing Systems with Finite QueuesTable 3Table 4Table 5Table 6Queue StationTechnician_1Technician_2Technician_3Technician_4Technician_5Technician_6Technician_7Technician_8Technician_9Technician_10Queue StationProcessing_1Processing_2Queue StationProc1Queue Stationrho1_1rho1_2rho1_3rho1_4rho1_5Queue StationTel_zero_1Tel_zero_2Tel_zero_3Arrival Rate1.10000002381.10000002381.10000002381.10000002381.10000002381.10000002381.10000002381.10000002381.10000002381.1000000238Arrival Rate1.51.5Arrival Rate1.5Arrival Rate510152025Arrival Rate101010Service Rate/Channel1111111111Service Rate/Channel22Service Rate/Channel2Service Rate/Channel55555Service Rate/Channel111Number of Servers1111111111Number of Servers12Number of Servers1Number of Servers12345Number of Servers101112Max. Number in System12345678910Max. Number in System45Max. Number in System4Max. Number in System55555Max. Number in System101112Number in Population******************************Number in Population******Number in Population***Number in Population***************Number in Population*********TypeM/M/1/1M/M/1/2M/M/1/3M/M/1/4M/M/1/5M/M/1/6M/M/1/7M/M/1/8M/M/1/9M/M/1/10TypeM/M/1/4M/M/2/5TypeM/M/1/4Q(n)Cum Pois.ProductTypeM/M/1/5M/M/2/5M/M/3/5M/M/4/5M/M/5/5TypeM/M/10/10M/M/11/11M/M/12/12Mean Time in Queue00.52380942591.06344408961.6188322962.18987443172.77644289283.37838518863.9955215574.62764945665.274539295Mean Number at Station1.44430220130.8494665623P(0)0.32778489590.365714291600Mean Number at Station2.52.727272748933.28888869293.5756611824Mean Number in Service7.85417652138.36767673498.80260849Prob. System Full0.52380949260.36555889250.28679162260.23981592060.20873402060.18673227730.17040365930.15785504880.14795039590.1399664879Mean Time at Station1.07428574560.5701705217P(1)0.24583867190.27428571870.13533528150.0371205349Mean Time at Station0.60000002380.33333334330.25142857430.21553398670.200000003Mean Time in Service111Throughput Rate1.34443020821.489846468P(2)0.18437900390.2057142890.40600585940.0835212067Mean Number in Queue1.66666671631.09090912340.61363635960.23703704770Throughput Rate7.85417652138.36767673498.80260849Efficiency0.67221510410.372461617P(3)0.1382842530.15428571680.67667639260.1044015022Mean Time in Queue0.40000002720.13333334010.05142857180.01553398230Efficiency0.7854176760.76069790120.7335507274Prob. System Full0.10371319950.0067689912P(4)0.103713189700.85712343450Throughput Rate4.16666650778.181818008411.931818008415.259258270317.8783054352Prob. System Full0.21458235380.16323235630.1197391376Critical Wait Time11time1Sum0.2250432439Efficiency0.83333331350.81818181280.79545456170.76296293740.7151322365P(Wait >= Critical Wait)0.2250432670.0111323912Prob. All Servers Busy0.83333337310.72727274890.61363637450.47407409550.2848677933Random VariableProcServ_1ProcServ_2ProcServ_3ProcServ_4ProcServ_5Prob. System Full0.16666667160.18181818720.20454545320.23703704770.2848677933DistributionPoissonPoissonPoissonPoissonPoissonP(0)0.16666667160.09090909360.04545454680.02222222280.0109389238mean (theta)22222P(1)0.16666667160.18181818720.13636364040.0888888910.054694619c00000P(2)0.16666667160.18181818720.20454546060.17777778210.1367365476d-10123P(3)0.16666667160.18181818720.20454546060.23703704280.227894246P(c