pertemuan 17 queueing models
DESCRIPTION
Pertemuan 17 QUEUEING MODELS. Matakuliah: D0174/ Pemodelan Sistem dan Simulasi Tahun: Tahun 2009. Learning Objectives. Terminologi Model Antrian Struktur Dasar Model Antrian Implementasi model antrian pada single station dan networks. Struktur Dasar Model Antrian. - PowerPoint PPT PresentationTRANSCRIPT
![Page 1: Pertemuan 17 QUEUEING MODELS](https://reader036.vdocument.in/reader036/viewer/2022062321/568135e4550346895d9d5810/html5/thumbnails/1.jpg)
![Page 2: Pertemuan 17 QUEUEING MODELS](https://reader036.vdocument.in/reader036/viewer/2022062321/568135e4550346895d9d5810/html5/thumbnails/2.jpg)
Pertemuan 17
QUEUEING MODELS
Matakuliah : D0174/ Pemodelan Sistem dan Simulasi
Tahun : Tahun 2009
![Page 3: Pertemuan 17 QUEUEING MODELS](https://reader036.vdocument.in/reader036/viewer/2022062321/568135e4550346895d9d5810/html5/thumbnails/3.jpg)
Learning Objectives
• Terminologi Model Antrian• Struktur Dasar Model Antrian• Implementasi model antrian pada single station
dan networks
![Page 4: Pertemuan 17 QUEUEING MODELS](https://reader036.vdocument.in/reader036/viewer/2022062321/568135e4550346895d9d5810/html5/thumbnails/4.jpg)
Struktur Dasar Model Antrian
![Page 5: Pertemuan 17 QUEUEING MODELS](https://reader036.vdocument.in/reader036/viewer/2022062321/568135e4550346895d9d5810/html5/thumbnails/5.jpg)
Struktur Dasar Model Antrian
![Page 6: Pertemuan 17 QUEUEING MODELS](https://reader036.vdocument.in/reader036/viewer/2022062321/568135e4550346895d9d5810/html5/thumbnails/6.jpg)
Struktur Dasar Model Antrian
![Page 7: Pertemuan 17 QUEUEING MODELS](https://reader036.vdocument.in/reader036/viewer/2022062321/568135e4550346895d9d5810/html5/thumbnails/7.jpg)
![Page 8: Pertemuan 17 QUEUEING MODELS](https://reader036.vdocument.in/reader036/viewer/2022062321/568135e4550346895d9d5810/html5/thumbnails/8.jpg)
![Page 9: Pertemuan 17 QUEUEING MODELS](https://reader036.vdocument.in/reader036/viewer/2022062321/568135e4550346895d9d5810/html5/thumbnails/9.jpg)
![Page 10: Pertemuan 17 QUEUEING MODELS](https://reader036.vdocument.in/reader036/viewer/2022062321/568135e4550346895d9d5810/html5/thumbnails/10.jpg)
![Page 11: Pertemuan 17 QUEUEING MODELS](https://reader036.vdocument.in/reader036/viewer/2022062321/568135e4550346895d9d5810/html5/thumbnails/11.jpg)
![Page 12: Pertemuan 17 QUEUEING MODELS](https://reader036.vdocument.in/reader036/viewer/2022062321/568135e4550346895d9d5810/html5/thumbnails/12.jpg)
![Page 13: Pertemuan 17 QUEUEING MODELS](https://reader036.vdocument.in/reader036/viewer/2022062321/568135e4550346895d9d5810/html5/thumbnails/13.jpg)
![Page 14: Pertemuan 17 QUEUEING MODELS](https://reader036.vdocument.in/reader036/viewer/2022062321/568135e4550346895d9d5810/html5/thumbnails/14.jpg)
SINGLE WORKSTATION• SYSTEM: STATION + INPUT QUEUE• INPUT: Batches of raw materials.• WORKSTATION: one or more identically capable
processors (servers).• OUTPUT: Completed products.• SIMPLEST SPECIAL CASE (M/M/1):
– Batch size = 1 ; Server size = 1– Exponential intearrival and service times– FCFS service policy – Service time = set-up time + processing time
![Page 15: Pertemuan 17 QUEUEING MODELS](https://reader036.vdocument.in/reader036/viewer/2022062321/568135e4550346895d9d5810/html5/thumbnails/15.jpg)
Single Station (cont’d)• Average arrival rate: • Average service rate: • Utilization factor (expected number of items in
process): = / • Expected number of items at station: L = Lq + • Expected throughput time: W = Wq + 1/• Actual number of items at station: n• Probability of having n items at time t: pt(n)
![Page 16: Pertemuan 17 QUEUEING MODELS](https://reader036.vdocument.in/reader036/viewer/2022062321/568135e4550346895d9d5810/html5/thumbnails/16.jpg)
Single Station (cont’d)• Probability of n = 0 at t
pt+t(0) = pt(0) (1 - t) + pt(1) t• Probability of n > 0 at t
pt+t(n) = pt(n) (1 - t - t) + pt(n+1) t + pt(n-1) t
![Page 17: Pertemuan 17 QUEUEING MODELS](https://reader036.vdocument.in/reader036/viewer/2022062321/568135e4550346895d9d5810/html5/thumbnails/17.jpg)
Single Station (cont’d)• In rate form:• For n = 0
dpt+t(0)/dt = - pt(0) + pt(1)• For n > 0
dpt+t(n)/dt = - ( + ) pt(n) +
pt(n+1) + pt(n-1)
![Page 18: Pertemuan 17 QUEUEING MODELS](https://reader036.vdocument.in/reader036/viewer/2022062321/568135e4550346895d9d5810/html5/thumbnails/18.jpg)
Single Station (cont’d)• At steady state pt+t(n) = pt(n) = p(n) :• For n = 0
p(0) = p(1)• For n > 0
( + ) p(n) = p(n+1) + p(n-1)
![Page 19: Pertemuan 17 QUEUEING MODELS](https://reader036.vdocument.in/reader036/viewer/2022062321/568135e4550346895d9d5810/html5/thumbnails/19.jpg)
Single Station (cont’d)• Steady state probabilities:• For n = 0
p(1) = p(0)• For n > 0
p(n+1) = [( + )/] p(n) - p(n-1)
![Page 20: Pertemuan 17 QUEUEING MODELS](https://reader036.vdocument.in/reader036/viewer/2022062321/568135e4550346895d9d5810/html5/thumbnails/20.jpg)
Single Station (cont’d)• Steady state probabilities (cont’d):
p(n) = n p(0)• Constraint:
p(n) = 1
![Page 21: Pertemuan 17 QUEUEING MODELS](https://reader036.vdocument.in/reader036/viewer/2022062321/568135e4550346895d9d5810/html5/thumbnails/21.jpg)
Single Station (cont’d)• Combining:
p(0) = • Also:
p(n) = n
• Expected number of items in system
L = n p(n) = /
![Page 22: Pertemuan 17 QUEUEING MODELS](https://reader036.vdocument.in/reader036/viewer/2022062321/568135e4550346895d9d5810/html5/thumbnails/22.jpg)
Single Station (cont’d)• Expected throughput time:
W = 1/ • Little’s Law:
L = W• See summary in Table 11.1, p. 366• See Example 11.1
![Page 23: Pertemuan 17 QUEUEING MODELS](https://reader036.vdocument.in/reader036/viewer/2022062321/568135e4550346895d9d5810/html5/thumbnails/23.jpg)
Single Station (cont’d)• Poisson arrivals, general FCFS service• M/G/1
E(S) = expectation for service time (1/)E(T) = expectation for throughput time TE(N) = expectation for number of jobs N
• See Example 11.2, p. 367
![Page 24: Pertemuan 17 QUEUEING MODELS](https://reader036.vdocument.in/reader036/viewer/2022062321/568135e4550346895d9d5810/html5/thumbnails/24.jpg)
Single Station (cont’d)• How about other that FCFS policy?• If multiple parts with different priorities are being
processed then priority service may have to be instituted
• See Sec. 11.2.3 and Example 11.3, p. 369
![Page 25: Pertemuan 17 QUEUEING MODELS](https://reader036.vdocument.in/reader036/viewer/2022062321/568135e4550346895d9d5810/html5/thumbnails/25.jpg)
Networks of Workstations
• Consider M workstations with jobs moving between workstation pairs following a routing scheme.
• If an external arrival process generates jobs that enter the network anytime, we have an open network.
• If the number of jobs in the network is maintained constant we have a closed network.
![Page 26: Pertemuan 17 QUEUEING MODELS](https://reader036.vdocument.in/reader036/viewer/2022062321/568135e4550346895d9d5810/html5/thumbnails/26.jpg)
Facts about Networks
• The sum of independent Poisson random variables is Poisson.
• If arrival rate is Poisson, the time interval between arrivals is Exponential.
• If service time is Exponential , the output rate is Poisson.
![Page 27: Pertemuan 17 QUEUEING MODELS](https://reader036.vdocument.in/reader036/viewer/2022062321/568135e4550346895d9d5810/html5/thumbnails/27.jpg)
Facts about Networks (cont’d)• The interdeparture time from an M/M/c
system with infinite queue capacity is Exponential.
• If a Poisson process of rate is split into multiple processes with probability pi, the individual streams become Poisson with arrival rates equal to pi
![Page 28: Pertemuan 17 QUEUEING MODELS](https://reader036.vdocument.in/reader036/viewer/2022062321/568135e4550346895d9d5810/html5/thumbnails/28.jpg)
Open Networks• Illustration of Facts:
– See Example 11.4, p. 372
• Poisson Arrivals and FCFS policy– Parts are taken from Warehouse for Kitting– Kits are sent to Assembly station(s)– Finished parts are sent to Inspection/Packing– See Fig. 11.2, p. 373
![Page 29: Pertemuan 17 QUEUEING MODELS](https://reader036.vdocument.in/reader036/viewer/2022062321/568135e4550346895d9d5810/html5/thumbnails/29.jpg)
Open Networks (cont’d)• Kitting-Assembly-Inspect/Pack Problem
– Kitting queue has always 1 hr worth of work– Kitting rate = 10 kits/hr– Assembly rate = 12 parts/hr– Inspection/Pack rate = 15 parts/hr– Assume all times are Exponential.– Serial System with Random Processing Times.
![Page 30: Pertemuan 17 QUEUEING MODELS](https://reader036.vdocument.in/reader036/viewer/2022062321/568135e4550346895d9d5810/html5/thumbnails/30.jpg)
Kitting-Assembly-Inspect/Pack• Output rate from Kitting is Poisson.• Arrival time into Assembly is Exponential.• Output from Assembly is Poisson.• Arrival time into Inspect/Pack is Exponential.• State of system described by number of jobs at
Assembly and Inspect/Pack (n1, n2)
![Page 31: Pertemuan 17 QUEUEING MODELS](https://reader036.vdocument.in/reader036/viewer/2022062321/568135e4550346895d9d5810/html5/thumbnails/31.jpg)
Kitting-Assembly-Inspect/Pack• States and transitions diagram (Fig. 11.3)• Steady-state balance equations (Eqn. 11.13,
p. 373)• Product Form Solution
p(n1,n2) = (1 - 1) 1n1 (1 - 2) 2
n2
• Recall for single workstationp(n) = n
![Page 32: Pertemuan 17 QUEUEING MODELS](https://reader036.vdocument.in/reader036/viewer/2022062321/568135e4550346895d9d5810/html5/thumbnails/32.jpg)
Important Note• The product form solution allows the
analysis of the M-station network by first analyzing the M individual stations separatedly and then combining the results.
• See Example 11.5
![Page 33: Pertemuan 17 QUEUEING MODELS](https://reader036.vdocument.in/reader036/viewer/2022062321/568135e4550346895d9d5810/html5/thumbnails/33.jpg)
Jackson’s Generalization
• M workstations with cj servers each.
• External arrivals are Poisson with rate j
• FCFS• Service times are Exponential w/mean 1/j
• Job at station j transfers to k with probability pjk
• Queue sizes are unlimited.
![Page 34: Pertemuan 17 QUEUEING MODELS](https://reader036.vdocument.in/reader036/viewer/2022062321/568135e4550346895d9d5810/html5/thumbnails/34.jpg)
Jackson (cont’d)
• Effective arrival rate = External arrivals + Internal arrivals
j’ = j + k k’ pkj
• Note this is a system of linear algebraic equations for the various j’
• Utilization factors must then be computed using the Effective arrival rates.
![Page 35: Pertemuan 17 QUEUEING MODELS](https://reader036.vdocument.in/reader036/viewer/2022062321/568135e4550346895d9d5810/html5/thumbnails/35.jpg)
Jackson (cont’d)• The state of system is given by the vector
n = (n1, n2, n3, ..., nM)• The probability of the system being in a state n
is p(n) .
![Page 36: Pertemuan 17 QUEUEING MODELS](https://reader036.vdocument.in/reader036/viewer/2022062321/568135e4550346895d9d5810/html5/thumbnails/36.jpg)
Procedure for Open Networks
1.- Solve for the effective arrival rates in all workstations (Eqn. 11.15)
2.- Analyze each station independently using Table 11.1.
3.- Aggregate results across stations to obtain performance measures.
• See Example 11.6, p. 377, Ex. 11.7, p. 378
![Page 37: Pertemuan 17 QUEUEING MODELS](https://reader036.vdocument.in/reader036/viewer/2022062321/568135e4550346895d9d5810/html5/thumbnails/37.jpg)
Closed Networks• Sometimes it may be convenient not to
introduce new jobs into the system but until a unit is completed and delivered.
• This maintains the number of jobs in the system at a constant level N .
• In this case WIP becomes a control parameter not an output statistic.
![Page 38: Pertemuan 17 QUEUEING MODELS](https://reader036.vdocument.in/reader036/viewer/2022062321/568135e4550346895d9d5810/html5/thumbnails/38.jpg)
Closed Networks• As N increases, both peoduction rate and
throughput increase.• Production rate is limited by lowest service rate
station.• Worsktations are not independent now.• Set of possible states is such that
nj = N
![Page 39: Pertemuan 17 QUEUEING MODELS](https://reader036.vdocument.in/reader036/viewer/2022062321/568135e4550346895d9d5810/html5/thumbnails/39.jpg)
Mean Value Analysis
• Assume P part types ( njp = Np; Np = N)• Mean service time for part p on station j = 1/jp
• Throughput time of part p at j
Wjp = 1/jp + ((Np-1)/Np) Ljp/ jp +
Ljr/ jp
![Page 40: Pertemuan 17 QUEUEING MODELS](https://reader036.vdocument.in/reader036/viewer/2022062321/568135e4550346895d9d5810/html5/thumbnails/40.jpg)
MVA (cont’d)• Throughput rates
Xp = Np/( vjp Wjp)• Number of visits of part p to station j = vjp
• Queue lengths
Ljp = Xp vjp Wjp
![Page 41: Pertemuan 17 QUEUEING MODELS](https://reader036.vdocument.in/reader036/viewer/2022062321/568135e4550346895d9d5810/html5/thumbnails/41.jpg)
MVA (cont’d)• Iterative Solution Procedure1.- Guess the values of Ljp2.- Obtain Wjp3.- Compute Xp4.- Compute improved values of Ljp5.- Repeat until satisfied.• See Example 11.0, pp. 388-392
![Page 42: Pertemuan 17 QUEUEING MODELS](https://reader036.vdocument.in/reader036/viewer/2022062321/568135e4550346895d9d5810/html5/thumbnails/42.jpg)
Product Form Solutions forClosed Networks
• Probability of selecting part of type p to enter the system next dp
• Station visit count vj = vjp dp
• Total work required at station jj = vjp dp jp
• Service rate at j 1 jp = j / vj
![Page 43: Pertemuan 17 QUEUEING MODELS](https://reader036.vdocument.in/reader036/viewer/2022062321/568135e4550346895d9d5810/html5/thumbnails/43.jpg)
Product Form Solutions forClosed Networks (cont’d)
• Rate station j serves customers under nrj(n) = min(nj,cj) j
• Probability of job leaving station j for k pjk
• Steady state equation (Eqn 11.32, p. 394)
p(n) rj(n) = p(njk) pjk rj(njk) • See Example 11.10, p. 394-
![Page 44: Pertemuan 17 QUEUEING MODELS](https://reader036.vdocument.in/reader036/viewer/2022062321/568135e4550346895d9d5810/html5/thumbnails/44.jpg)
Product Form Solutions forClosed Networks (cont’d)
• The solution to the balance equations is
p(n) = G-1 (N) (f1*f2*f3 ...fM)• Where, if nj < cj
fj(nj) = j nj/nj!
• And if nj > cjfj(nj) = j nj/(cj! cjnj-cj)
• And G-1 (N) = (f1*f2*f3 ...fM)
![Page 45: Pertemuan 17 QUEUEING MODELS](https://reader036.vdocument.in/reader036/viewer/2022062321/568135e4550346895d9d5810/html5/thumbnails/45.jpg)
Tugas
1. Di Stasiun Pengisian Bahan Bakar Umum (SPBU), sering terjai antrian. Buatlah suatu studi kasus di SPBU terdekat dengan mengimplementasikan model antrian
2. Jelaskan manfaat dari implementasi model antrian !
3. Jelaskan yang dimaksud dengan Model Transportasi !
![Page 46: Pertemuan 17 QUEUEING MODELS](https://reader036.vdocument.in/reader036/viewer/2022062321/568135e4550346895d9d5810/html5/thumbnails/46.jpg)
Daftar Pustaka
Harrel. Ghosh. Bowden. (2000). Simulation Using Promodel. McGraw-Hill. New York.
RG Coyle. (1996). System Dynamics Modelling : A Practice Approach. Chapman & Hall. United Kingdom.
![Page 47: Pertemuan 17 QUEUEING MODELS](https://reader036.vdocument.in/reader036/viewer/2022062321/568135e4550346895d9d5810/html5/thumbnails/47.jpg)
TERIMA KASIH