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Modeling and Analysis of Manufacturing

SystemsSession 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 (contd)

- 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 (contd)

- Probability of n 0 at t
- pt?t(0) pt(0) (1 - ? ?t) pt(1) ? ?t
- Probability of n gt 0 at t
- pt?t(n) pt(n) (1 - ? ?t - ? ?t) pt(n1) ?

?t pt(n-1) ? ?t

Single Station (contd)

- In rate form
- For n 0
- dpt?t(0)/dt - ? pt(0) ? pt(1)
- For n gt 0
- dpt?t(n)/dt - (? ? ) pt(n)
- ? pt(n1) ? pt(n-1)

Single Station (contd)

- At steady state pt?t(n) pt(n) p(n)
- For n 0
- ? p(0) ? p(1)
- For n gt 0
- (? ? ) p(n) ? p(n1) ? p(n-1)

Single Station (contd)

- Steady state probabilities
- For n 0
- p(1) ??????? p(0)
- For n gt 0
- p(n1) (? ? )/? p(n) -
- ??????? p(n-1)

Single Station (contd)

- Steady state probabilities (contd)
- p(n) ???????n p(0)
- Constraint
- ? p(n) 1

Single Station (contd)

- Combining
- p(0) ???????
- Also
- p(n) ?????????n
- Expected number of items in system
- L ? n p(n) ? /???????

Single Station (contd)

- Expected throughput time
- W 1/? ???????
- Littles Law
- L ?W
- See summary in Table 11.1, p. 366
- See Example 11.1

Single Station (contd)

- 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 (contd)

- 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 (contd)

- 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 (contd)

- 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) ?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

Jacksons 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 (contd)

- 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 (contd)

- 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 (contd)

- Throughput rates
- Xp Np/(? vjp Wjp)
- Number of visits of part p to station j vjp
- Queue lengths
- Ljp Xp vjp Wjp

MVA (contd)

- 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

(contd)

- 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 forClosed Networks

(contd)

- The solution to the balance equations is
- p(n) G-1 (N) (f1f2f3 ...fM)
- Where, if nj lt cj
- fj(nj) ?j nj/nj!
- And if nj gt cj
- fj(nj) ?j nj/(cj! cjnj-cj)
- And
- G-1 (N) ? (f1f2f3 ...fM)

Hybrid Systems

- See Sec. 11.5