modeling and analysis of manufacturing systems session 2 queueing models january 2001

Modeling and Analysis of Manufacturing Systems

Session 2QUEUEING MODELS

January 2001

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


• 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)


• 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)


• Steady state probabilities:

• For n = 0

p(1) = p(0)• For n > 0

p(n+1) = [( + )/] p(n) -

p(n-1)


• Steady state probabilities (cont’d):

p(n) = n p(0)• Constraint:

p(n) = 1


• Combining:

p(0) = • Also:

p(n) = n

• Expected number of items in system

L = n p(n) = /


• Expected throughput time:

W = 1/ • Little’s Law:

L = W• See summary in Table 11.1, p. 366

• See Example 11.1


• 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


• 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 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

• 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 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 j

j = 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 > cj

fj(nj) = j nj/(cj! cjnj-cj)

• And

G-1 (N) = (f1*f2*f3 ...fM)

Hybrid Systems

• See Sec. 11.5

modeling and analysis of manufacturing systems session 2 queueing models january 2001

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