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

Struktur Dasar Model Antrian

Struktur Dasar Model Antrian

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

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 TE(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– 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

– 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• 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) 2

n2

• Recall for single workstationp(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

• 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 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

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

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-

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)

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