Transcript QUEUEING MODELS

Modeling and Analysis of
Manufacturing Systems
Session 2
QUEUEING 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
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
–
–
–
–
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) 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
•
•
•
•
•
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)
Hybrid Systems
• See Sec. 11.5