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# Queue Calculations

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Queuing Computations

Intro

QUEUING TEMPLATES 1995 by David W. AshleyRevised May 21, 1997

Modified for class use by Yong-Pin Zhou, 2006

Kendall Notation

This worksheet computes queuing results for the following models:

M / G / 1

M / M / m

M / M / m / k - M / M / m with finite queue length

G / G / m *

Click on the page tab to use the model of your choice. Enter the required parameters in the boxes.

Queueing models are classified using a system called Kendall notation. The general format is */*/m/k, where the first character denotes the assumptions made about the arrival process. M means Poisson, D means deterministic (no randomness), and G means generalno assumptions are necessary about the arrival process. The second character denotes assumptions made about the service process. The third character, m, is the number of servers in the queueing system. The last character, k, is the number of total waiting space in the queueing system (which includes m). When the waiting space is infinite, k is often omitted.* This model allows general arrival and service time distribubtions. The formulas used in the spreadsheet is approximate, which works best when utilization is high. When using this spreadsheet, make sure that utilization is at least 0.5. We will not have time to go through this in class, but it may be useful to you later so I have chosen to include it here.

MG1M/G/1/ queueing computationsArrival rate (l)0.025per minuteServicesvc time avg (1/m) 24.0minutesvc time std dev (s)24.0minuteservice rate (m)0.041666666666666664per minuteTime Unitminute

Utilization0.6000000000000001Lq, expected queue length0.900000Ls, expected number in system1.500000Wq, expected time in queue36.000000minutesWs, expected total time in system60.000000minutes

l1/msmWs0.02524.0100.00.041666666666666664354.50.02524.060.00.041666666666666664154.50.02524.030.00.04166666666666666470.1250.02524.024.00.04166666666666666460.00.02524.015.00.04166666666666666449.030.02524.08.00.04166666666666666444.00.02524.01.00.04166666666666666442.030.02524.00.00.04166666666666666442.0

MMmM/M/m/ queueing computationslambda/mu2.0m-13.0THE ARRIVAL RATE SHOULD BE LESS THAN THE OVERALL SERVICE RATE!Arrival rate (l)2.0per minuteAssumes Poisson process for/m0.5 Service rate (m)1.0per minutearrivals and services.1.3333333333333333 m factorial =24# of servers (m)4.0 (max of 40)Time UnitminuteUtilization50.00%p(0) =0.130434782608695650.17391304347826086p0, probability that the system is empty0.130435p(0)++p(n)np(n)Lq, expected queue length0.1739130.130434782608695650.01.00.130434782608695650.13043478260869565Ls, expected number in system2.1739130.39130434782608691.02.00.26086956521739130.2608695652173913Wq, expected time in queue0.086957minutes0.65217391304347832.02.00.26086956521739130.2608695652173913Ws, expected total time in system1.086957minutes0.82608695652173913.01.33333333333333330.173913043478260860.17391304347826086Probability that a customer has to wait in queue0.1739130.91304347826086964.00.00.086956521739130430.00.95652173913043485.00.00.0434782608695652160.00.97826086956521746.00.00.0217391304347826080.00.98913043478260887.00.00.0108695652173913040.00.99456521739130448.00.00.0054347826086956520.00.99728260869565229.00.00.0027173913043478260.00.998641304347826210.00.00.0013586956521739130.00.999320652173913111.00.06.793478260869565E-40.00.999660326086956512.00.03.3967391304347825E-40.00.999830163043478313.00.01.6983695652173913E-40.0t =5.00.999915081521739114.00.08.491847826086956E-50.00.999957540760869615.00.04.245923913043478E-50.0 Prob(waiting in queue > t) =Prob(wait) * e-(mm-l)t0.999978770380434816.00.02.122961956521739E-50.0 =7.895639958693018E-60.999989385190217417.00.01.0614809782608695E-50.00.999994692595108818.00.05.307404891304348E-60.00.999997346297554419.00.02.653702445652174E-60.00.999998673148777220.00.01.326851222826087E-60.00.999999336574388721.06.634256114130435E-70.00.999999668287194322.03.3171280570652173E-70.00.999999834143597223.01.6585640285326086E-70.00.999999917071798624.08.292820142663043E-80.00.999999958535899325.04.1464100713315216E-80.00.999999979267949726.02.0732050356657608E-80.00.999999989633974927.01.0366025178328804E-80.00.999999994816987428.05.183012589164402E-90.00.999999997408493829.02.591506294582201E-90.00.999999998704246930.01.2957531472911005E-90.00.999999999352123531.06.478765736455503E-100.00.999999999676061832.03.2393828682277513E-100.00.999999999838030933.01.6196914341138756E-100.00.999999999919015434.08.098457170569378E-110.00.999999999959507735.04.049228585284689E-110.00.999999999979753936.02.0246142926423445E-110.00.99999999998987737.01.0123071463211723E-110.00.999999999994938538.05.061535731605861E-120.00.999999999997469239.02.5307678658029307E-120.00.999999999998734740.01.2653839329014653E-120.0

MMmk-finite queue lengthM/M/m/k - M/M/m with Finite Queue Arrival rate (l)10.0per hour Service rate (m)6.0per hour Number of servers (m) 3.0 (max of 40) Maximum number in system (k)5.0 k>=m, (max of 40) Maximum queue length (k-m)2.0 (max of 40 combined)l/m1.66666666666666675.0 Time UnitshourUtilization53.15%/m0.5555555555555556p0, probability that the system is empty0.18200.7716049382716051 m factorial =6computation of Lprob waitLq, expected queue length0.1647comp of Lq0.99999999999999990.73818176774121950.8089663208122952Ls, expected number in system1.7591p0 =0.182017422182766433.00.16472007989790824nP(n)0.00161839652482557560.0301023537195132460.7381817677412195Wq, expected time in queue0.0172hours2.06.00.00.00.182017422182766430.182017422182766430.00.18201742218276643Ws, expected total time in system0.1839hours0.01.01.06.00.01.00.30336237030461070.30336237030461070.45504355545691610.30336237030461070.30336237030461070.3033623703046107Probability that a customer has to wait in queue0.2618181.01.66666666666666674.00.55555555555555561.06.00.02.00.25280197525384230.25280197525384230.25280197525384230.25280197525384230.50560395050768460.2528019752538423Probability that a customer is lost0.0433472.01.3888888888888895.00.3086419753086421.06.00.03.00.140445541807690170.140445541807690170.140445541807690170.00.00.0Rate of customer served9.56653.00.77160493827160516.00.01.06.00.078025301004272324.00.078025301004272320.058518975753204240.078025301004272320.00.00.0Rate of customer lost0.43354.00.07.00.01.06.00.086694778893635925.00.043347389446817960.019506325251068080.043347389446817960.00.00.05.00.08.00.01.06.00.06.00.00.0054184236808522450.024081883026009980.00.00.06.00.09.00.01.06.00.07.00.00.00129010087639339190.0133788239033388770.00.00.07.00.010.00.01.06.00.08.00.02.6877101591529E-40.0074326799462993780.00.00.08.00.011.00.01.06.00.09.00.04.977241035468333E-50.0041292666368329880.00.00.09.00.012.00.01.06.00.010.00.08.295401725780555E-60.00229403702046277130.00.00.010.00.013.00.01.06.00.011.00.01.25687904936069E-60.00127446501136820630.00.00.011.00.014.00.01.06.00.012.00.01.7456653463342917E-77.080361174267814E-40.00.00.012.00.015.00.01.06.00.013.00.02.238032495300374E-83.933533985704341E-40.00.00.013.00.016.00.01.06.00.014.00.02.664324399167112E-92.1852966587246344E-40.00.00.014.00.017.00.01.06.00.015.00.02.9603604435190136E-101.2140536992914636E-40.00.00.015.00.018.00.01.06.00.016.00.03.083708795332306E-116.744742773841465E-50.00.00.016.00.019.00.01.06.00.017.00.03.023243916992457E-123.747079318800813E-50.00.00.017.00.020.00.01.06.00.018.00.02.799299923141164E-132.081710732667119E-50.00.00.018.00.021.00.01.06.00.019.00.02.4555262483694418E-141.156505962592844E-50.00.00.019.00.022.00.01.06.00.020.00.02.0462718736412018E-156.4250331255158E-60.00.00.020.00.023.00.01.06.00.021.00.01.6240252965406364E-163.569462847508778E-60.00.00.021.00.024.00.01.06.00.022.00.01.2303221943489671E-171.983034915282655E-60.00.00.022.00.025.00.01.06.00.023.00.08.915378219920052E-191.1016860640459192E-60.00.00.023.00.026.00.01.06.00.024.00.06.191234874944481E-206.120478133588442E-70.00.00.024.00.027.00.01.06.00.025.00.04.127489916629654E-213.4002656297713565E-70.00.00.025.00.028.00.01.06.00.026.00.02.645826869634394E-221.889036460984087E-70.00.00.026.00.029.00.01.06.00.027.00.01.6332264627372802E-231.0494647005467152E-70.00.00.027.00.030.00.01.06.00.028.00.09.721586087721906E-255.830359447481752E-80.00.00.028.00.031.00.01.06.00.029.00.05.587118441219486E-263.2390885819343065E-80.00.00.029.00.032.00.01.06.00.030.00.03.1039546895663813E-271.7994936566301707E-80.00.00.030.00.033.00.01.06.00.031.00.01.6687928438528933E-289.997186981278726E-90.00.00.031.00.034.00.01.06.00.032.00.08.691629395067153E-305.553992767377069E-90.00.00.032.00.035.00.01.06.00.033.00.04.389711815690482E-313.0855515374317057E-90.00.00.033.00.036.00.01.06.00.034.00.02.1518195174953342E-321.7141952985731702E-90.00.00.034.00.037.00.01.06.00.035.00.01.0246759607120638E-339.52330721429539E-100.00.00.035.00.038.00.01.06.00.036.00.04.7438701884817773E-355.290726230164106E-100.00.00.036.00.039.00.01.06.00.037.00.02.13687846328008E-362.9392923500911697E-100.00.00.037.00.040.00.01.06.00.038.00.09.372273961754737E-381.632940194495095E-100.00.00.038.00.041.00.01.06.00.039.00.04.0052452828011693E-399.071889969417191E-110.00.00.039.00.042.00.01.06.00.040.00.01.668852201167154E-405.039938871898441E-110.00.00.040.00.043.00.01.06.01.06.01.06.01.06.01.06.01.06.01.06.01.06.0

0.0

GGmG/G/m queuing computationsArrival rate (l)150.0per hoursinter-arrival0.006666666666666667hint: for Poisson arrival, this = 1/lService rate (m)20.0per hoursservice0.05hint: for exponential service, this = 1/mNumber of servers 9.0 (max of 40)Time Unithour

Utilization0.833333Lq, expected queue length2.545743Ls, expected number in system10.045743Wq, expected time in queue0.016972hoursWs, expected total time in system0.066972hoursProbability that a customer has to wait in queue0.509149

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