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Chapter 12- Waiting Lines, Exercises of Econometrics and Mathematical Economics

Brock UniversityEconometrics and Mathematical Economics

Metodos Cuantitativos para los negocios

Typology: Exercises

2021/2022

Uploaded on 11/27/2023

james-guerrero
james-guerrero 🇨🇦

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Waiting Lines
Multiple server model (M/M/s)
Data
12
15
Number of servers(s) 2
Server cost $/time) 10
Waiting cost ($/time) 50
Computations
n or s (lam/mu)^n/n!
Cumsum(n-1)
0 1
1 0.8 1
2 0.32 1.8
3
0.085333333333
2.12
4
0.017066666667
2.20533333
5
0.002730666667
2.2224
6
0.000364088889
2.22513067
7 4.16101587E-05 2.22549476
8 4.16101587E-06 2.22553637
Arrival rate (l)
Service rate (m)
The arrival RATE and service RATE both must be rates and use the same time unit. Given a time
such as 10 minutes, convert it to a rate such as 6 per hour.
pf3
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pf1a
pf1b
pf1c
pf1d
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pf1f
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pf24
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Partial preview of the text

Download Chapter 12- Waiting Lines and more Exercises Econometrics and Mathematical Economics in PDF only on Docsity!

Waiting Lines Multiple server model (M/M/s)

Data 12 15 Number of servers(s) 2 Server cost $/time) 10 Waiting cost ($/time) 50 Computations n or s (lam/mu)^n/n! Cumsum(n-1) 0 1 1 0.8 1 2 0.32 1. 3 0.085333333333 2. 4 0.017066666667 2. 5 0.002730666667 2. 6 0.000364088889 2. 7 4.16101587E-05 2. 8 4.16101587E-06 2. Arrival rate (l) Service rate (m) The arrival RATE and service RATE both must be rates and use the same time unit. Given a time such as 10 minutes, convert it to a rate such as 6 per hour.

9 3.69868078E-07 2.

10 2.95894462E-08 2.

11 2.15195972E-09 2.

12 1.43463982E-10 2.

13 8.82855271E-12 2.

14 5.04488727E-13 2.

15 2.69060654E-14 2.

16 1.34530327E-15 2.

17 6.33083892E-17 2.

18 2.81370619E-18 2.

19 1.18471839E-19 2.

20 4.73887358E-21 2.

21 1.80528517E-22 2.

22 6.56467336E-24 2.

Sensitivity Table (Excel Data Table)

Numbe

Number of servers

Waiting Lines Multiple server model (M/M/s)

Data 3 4 Number of servers(s) 2 Server cost $/time) 12 Waiting cost ($/time) 20 Computations n or s (lam/mu)^n/n! Cumsum(n-1) 0 1 1 0.75 1 2 0.28125 1. 3 0.0703125 2. 4 0.01318359375 2. 5 0.001977539063 2. 6 0.000247192383 2. 7 2.64848982E-05 2. 8 2.4829592E-06 2. Arrival rate (l) Service rate (m) The arrival RATE and service RATE both must be rates and use the same time unit. Given a time such as 10 minutes, convert it to a rate such as 6 per hour.

ultiple server model (M/M/s)

Sensitivity Table (Excel Da Results 2 0.375 0. 0.12273 0. Average number of customers in the system(L) 0.87273 0. 0.04091 0. Average time in the system(W) 0.29091 0. 0.45455 0. Cost - based on waiting 26.4545 26. Cost - based on system 41.4545 41. Costo Costo por 1 servidores 12 Costo pot 2 Servidores 24 term2 P0(s) Rho(s) Lq(s) 3 0.25 0.75 2. 0.45 0.454545 0.375 0. 0.09375 0.470588 0.25 0. 0.0162259615384615 0.472191 0.1875 0. 0.00232651654411765 0.47235 0.15 0. 0.000282505580357143 0.472365 0.125 1.906369E- 0.0000296630859375 0.472366 0.107142857142857 1.681422E- 2.73981705087746E-06 0.472367 0.09375 1.338825E- Average server utilization( r ) Average number of customers in the queue(Lq) Average waiting time in the queue(Wq) Probability (% of time) system is empty (P 0 ) El ahorro en costos por hora es sustancial, reduce de 360 a 19.64 si se agrega un cargador más, esto se debe a que los camiones en espera son muchos menos, por lo que los costos de espera se ven reducidos. Disminuyen de 2.25 camiones en espera a solo .1227. st be rates and use the same time unit. Given a time as 6 per hour.

9 4.04127474E-10 1.

10 1.51547803E-11 1.

11 5.16640237E-13 1.

12 1.61450074E-14 1.

13 4.65721368E-16 1.

14 1.24746795E-17 1.

15 3.11866987E-19 1.

16 7.30938251E-21 1.

17 1.61236379E-22 1.

18 3.35909123E-24 1.

19 6.62978532E-26 1.

20 1.24308475E-27 1.

21 2.21979419E-29 1.

22 3.7837401E-31 1.

ultiple server model (M/M/s)

Sensitivity Table (Excel Data Table) Results 1 1 0.375 0. 0.225 0. Average number of customers in the system(L) 0.6 0.6 0. 0.075 0.075 0. Average time in the system(W) 0.2 0.2 0. 0.625 0.625 0. term2 P0(s) Rho(s) Lq(s) L(s) 0.6 0.625 0.375 0.225 0. 0.0865384615384615 0.684211 0.1875 0.013663968 0. 0.0100446428571429 0.687117 0.125 0.000985977 0. 0.000909213362068966 0.68728 0.09375 6.46432E-05 0. 6.68087521114865E-05 0.687289 0.075 3.72299E-06 0. 4.119873046875E-06 0.687289 0.0625 1.8877E-07 0. 2.18625338572376E-07 0.687289 0.053571429 8.50522E-09 0. 1.01760623047447E-08 0.687289 0.046875 3.43962E-10 0. Average server utilization( r ) Average number of customers in the queue(Lq) Average waiting time in the queue(Wq) Probability (% of time) system is empty (P 0 ) ust be rates and use the same time unit. Given a time as 6 per hour.

e (Excel Data Table) Number of servers 2 3 4 5 6 7 8 0.388663968 0.375985977 0.375064643 0.375003723 0.375000189 0.375000009 0. 0.004554656 0.000328659 2.15477E-05 1.241E-06 6.29232E-08 2.83507E-09 1.14654E- 0.129554656 0.125328659 0.125021548 0.125001241 0.125000063 0.125000003 0. 0.684210526 0.687116564 0.687280133 0.687288839 0.68728926 0.687289278 0. Wq(s) W(S) 0.075 0. 0.004554656 0. 0.000328659 0. 2.15477E-05 0. 1.241E-06 0. 6.29232E-08 0. 2.83507E-09 0. 1.14654E-10 0.

  • 4.0595276809911E-07 0.
  • 3.21624415305575E-08 0.
  • 2.32074087906768E-09 0.
  • 1.53711408873314E-10 0.
  • 9.40747420426838E-12 0.
  • 5.35063800895451E-13 0.
  • 2.84219000878067E-14 0.
  • 1.41610870612931E-15 0.
  • 6.64347294233506E-17 0.
  • 2.94457624256711E-18 0.
  • 1.23679392850194E-19 0.
  • 4.93632664445949E-21 0.
  • 1.87678161531925E-22 0.
  • 6.81239687860716E-24 0. - 0. - 0. - 0.952380952 4 0.952380952 0. - 0.012698413 0.266666667 0.012698413 0. - 0.079365079 0.333333333 0.079365079 0. - 0.428571429 0.2 0.428571429 0. - 27.61904762 170 27.61904762 30. - 67.61904762 210 67.61904762 70. - Horas al día
  • Costo por 1 servidores 10 80 25.60 $ 1,280. Costo Costo * Día Wq Costo total
  • Costo pot 2 Servidores 20 160 1.22 $ 60. - 0.8 3.2 4 0.266666667 0. Rho(s) Lq(s) L(s) Wq(s) W(S) - 0.4 0.152380952 0.952380952 0.012698413 0.
    • 0.266666666666667 0.018920916 0.818920916 0.001576743 0. - 0.2 0.00239521 0.80239521 0.000199601 0.
      • 0.16 0.00027821 0.80027821 2.318418E-05 0.
    • 0.133333333333333 2.904053E-05 0.800029041 2.420044E-06 0.
    • 0.114285714285714 2.723756E-06 0.800002724 2.269797E-07 0. - 0.1 2.308228E-07 0.800000231 1.923524E-08 0.
  • 0.0888888888888889 1.779574E-08 0.800000018 1.482978E-09 0.
    • 0.08 1.256654E-09 0.800000001 1.047211E-10 0.
  • 0.0727272727272727 8.178636E-11 0.8 6.81553E-12 0.
  • 0.0666666666666667 4.933356E-12 0.8 4.11113E-13 0.
  • 0.0615384615384615 2.771836E-13 0.8 2.309864E-14 0.
  • 0.0571428571428571 1.457089E-14 0.8 1.214241E-15 0.
  • 0.0533333333333333 7.194807E-16 0.8 5.995673E-17 0.
    • 0.05 3.34894E-17 0.8 2.790784E-18 0.
  • 0.0470588235294118 1.474126E-18 0.8 1.228438E-19 0.
  • 0.0444444444444444 6.153876E-20 0.8 5.12823E-21 0.
  • 0.0421052631578947 2.442758E-21 0.8 2.035631E-22 0.
    • 0.04 9.241811E-23 0.8 7.701509E-24 0.
  • 0.0380952380952381 3.339772E-24 0.8 2.783143E-25 0.
  • 0.0363636363636364 1.155097E-25 0.8 9.625809E-27 0. - 0.80239521 0.80027821 0.800029041 0.800002724 0.800000231 0.800000018 0.
  • 0.000199601 2.318418E-05 2.42004E-06 2.269797E-07 1.923524E-08 1.482978E-09 1.047211E-
  • 0.066866267 0.066689851 0.066669087 0.066666894 0.066666686 0.066666668 0.
  • 0.449101796 0.449306784 0.449326977 0.449328801 0.449328952 0.449328963 0.
  • 40.11976048 50.01391051 60.00145203 70.00013619 80.00001154 90.00000089 100.
  • 80.11976048 90.01391051 100.001452 110.0001362 120.0000115 130.0000009 140.
  • $ 1,360.
    • $ 220.
    • 2.257235638507E-07 0.472367 0.0833333333333333 9.693115E-
  • 1.67767513672818E-08 0.472367 0.075 6.425494E-
  • 1.13549963522456E-09 0.472367 0.0681818181818182 3.924674E-
  • 7.05386137033438E-11 0.472367 0.0625 2.221339E-
  • 4.04877247021744E-12 0.472367 0.0576923076923077 1.170921E-
  • 2.15954119420161E-13 0.472367 0.0535714285714286 5.774123E-
  • 1.07571130914178E-14 0.472367 0.05 2.674369E-
  • 5.02586431320339E-16 0.472367 0.046875 1.167566E-
  • 2.21089704162553E-17 0.472367 0.0441176470588235 4.820095E-
  • 9.18851072286825E-19 0.472367 0.0416666666666667 1.887107E-
  • 3.61876278469126E-20 0.472367 0.0394736842105263 7.02486E-
  • 1.35425333193333E-21 0.472367 0.0375 2.492353E-
  • 4.82766234068827E-23 0.472367 0.0357142857142857 8.446023E-
  • 1.64302793947794E-24 0.472367 0.0340909090909091 2.739217E- - 3 0.872727273 0.764705882 0.751768096 0.750193929 0.750019064 0.
    • 0.75 0.040909091 0.004901961 0.000589365 6.4643E-05 6.354562E-06 5.604739E- - 1 0.290909091 0.254901961 0.250589365 0.250064643 0.250006355 0.
    • 0.25 0.454545455 0.470588235 0.472190692 0.472350355 0.472365187 0.
      • 57 26.45454545 36.29411765 48.03536191 60.00387858 72.00038127 84.
      • 72 41.45454545 51.29411765 63.03536191 75.00387858 87.00038127 99. - Horas al día - 18.00 $ 360.00 $ 360. Wq Costo total - 0.98 $ 19.64 $ 19.
        • 3 0.75 L(s) Wq(s) W(S)
  • 0.872727273 0.040909091 0.
  • 0.764705882 0.004901961 0.
  • 0.751768096 0.000589365 0.
  • 0.750193929 6.4643E-05 0.
  • 0.750019064 6.354562E-06 0.
  • 0.750001681 5.604739E-07 0.
  • 0.750000134 4.462751E-08 0.
    • 0.75000001 3.231038E-09 0.
  • 0.750000001 2.141831E-10 0. - 0.75 1.308225E-11 0. - 0.75 7.404463E-13 0. - 0.75 3.903071E-14 0. - 0.75 1.924708E-15 0. - 0.75 8.914562E-17 0. - 0.75 3.891886E-18 0. - 0.75 1.606698E-19 0. - 0.75 6.290355E-21 0. - 0.75 2.34162E-22 0. - 0.75 8.307844E-24 0. - 0.75 2.815341E-25 0. - 0.75 9.130723E-27 0.
    • 0.750000134 0.75000001 0.
  • 4.462751E-08 3.231038E-09 2.14183E-
    • 0.250000045 0.250000003 0.
    • 0.472366545 0.472366552 0.
    • 96.00000268 108.0000002
    • 111.0000027 123.0000002
  • 4.21698234096077E-10 0.687289 0.041666667 1.26012E-11 0.
  • 1.57452262730678E-11 0.687289 0.0375 4.21618E-13 0.
  • 5.34874598393921E-13 0.687289 0.034090909 1.29746E-14 0.
  • 1.66658140995173E-14 0.687289 0.03125 3.69491E-16 0.
  • 4.79554675512102E-16 0.687289 0.028846154 9.78988E-18 0.
  • 1.28180192907698E-17 0.687289 0.026785714 2.42468E-19 0.
  • 3.19863576624064E-19 0.687289 0.025 5.63689E-21 0.
    • 7.4848076930031E-21 0.687289 0.0234375 1.23461E-22 0.
  • 1.64873289760113E-22 0.687289 0.022058824 2.55599E-24 0.
  • 3.43056125439534E-24 0.687289 0.020833333 5.01657E-26 0.
  • 6.76327092938678E-26 0.687289 0.019736842 9.35904E-28 0.
  • 1.26683795903281E-27 0.687289 0.01875 1.66373E-29 0.
  • 2.26015408600171E-29 0.687289 0.017857143 2.82433E-31 0.
  • 3.84935408949507E-31 0.687289 0.017045455 4.58778E-33 0.
  • 4.20042E-12 0.
  • 1.40539E-13 0.
  • 4.32487E-15 0.
  • 1.23164E-16 0.
  • 3.26329E-18 0.
  • 8.08228E-20 0.
  • 1.87896E-21 0.
  • 4.11538E-23 0.
  • 8.51997E-25 0.
  • 1.67219E-26 0.
  • 3.11968E-28 0.
  • 5.54576E-30 0.
  • 9.41442E-32 0.
  • 1.52926E-33 0.