Matakuliah
Tahun
: D0174/ Pemodelan Sistem dan Simulasi
: Tahun 2009
Pertemuan 17
QUEUEING MODELS
Learning Objectives
• Terminologi Model Antrian
• Struktur Dasar Model Antrian
• Implementasi model antrian pada single station dan
networks
Struktur Dasar Model Antrian
Struktur Dasar Model Antrian
Struktur Dasar Model Antrian
SINGLE WORKSTATION
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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
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)
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
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)
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)
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)
Single Station (cont’d)
• Steady state probabilities (cont’d):
p(n) = n p(0)
• Constraint:
 p(n) = 1
Single Station (cont’d)
• Combining:
p(0) = 
• Also:
p(n) = n
• Expected number of items in system
L =  n p(n) =  /
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
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 T
E(N) = expectation for number of jobs N
• See Example 11.2, p. 367
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
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.
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.
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
Open Networks
• Illustration of Facts:
– See Example 11.4, p. 372
• Poisson Arrivals and FCFS policy
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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
Open Networks (cont’d)
• Kitting-Assembly-Inspect/Pack Problem
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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.
Kitting-Assembly-Inspect/Pack
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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)
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) 2n2
• Recall for single workstation
p(n) = n
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
Jackson’s Generalization
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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.
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.
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) .
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
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.
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
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
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
MVA (cont’d)
• Iterative Solution Procedure
1.- Guess the values of Ljp
2.- Obtain Wjp
3.- Compute Xp
4.- Compute improved values of Ljp
5.- Repeat until satisfied.
• See Example 11.0, pp. 388-392
Product Form Solutions for
Closed 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 j
j =  vjp dp jp
• Service rate at j
1 jp = j / vj
Product Form Solutions for
Closed Networks (cont’d)
• Rate station j serves customers under n
rj(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-
Product Form Solutions for
Closed 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 > cj
fj(nj) = j nj/(cj! cjnj-cj)
• And
G-1 (N) =  (f1*f2*f3 ...fM)
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 !
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.
TERIMA KASIH