Chapter%208%20Operations%20Scheduling

1
Chapter 8Operations Scheduling
2
Scheduling Problems in Operations

• Job Shop Scheduling
• Personnel Scheduling
• Facilities Scheduling
• Vehicle Scheduling and Routing
• Project Management
• Dynamic versus Static Scheduling

3
The Hierarchy of Production Decisions
4
Characteristics of the Job Shop Scheduling
Problem

• Job Arrival Pattern
• Number and Variety of Machines
• Number and Skill Level of Workers
• Flow Patterns
• Evaluation of Alternative Rules

5
Objectives in Job Shop Scheduling

• Meet due dates
• Minimize work-in-process (WIP) inventory
• Minimize average flow time
• Maximize machine/worker utilization
• Reduce set-up times for changeovers
• Minimize direct production and labor costs
• (note that these objectives can be conflicting)

6
Terminology

• Flow shop shop design in which machines are
  arranged in series
• A Pure Flow Shop
• In general flow shop a job may skip a particular
  machine

7
Terminology

• Job shop the sequencing of jobs through machines
  
• A job shop does not have the same restriction on
  workflow as a flow shop. In a job shop, jobs can
  be processed on machines in any order
• Usual job shop contains m machines and n jobs to
  be processed
• Each job requires m operations (one on each
  machine) in a specific order, but the order can
  be different for each job
• Real job shops might not require to use all m
  machines and yet may have to visit some machines
  more than once
• Workflow is not unidirectional in a job shop
• One Machine in a Job Shop

8
Terminology

• Parallel processing vs. sequential processing
  parallel processing means that the machines are
  identical
• In practice, there are often multiple copies of
  the same machine
• A job arriving at a work center can be scheduled
  on any one of a number machines ? more
  flexibility, complicating the scheduling problem
  further
• A factory might have multiple identical
  machines, purchased from the same manufacturer,
  that produce parts with higher quality on one
  machine than on any other
• Schedule provides the order in which jobs are to
  be executed, and projects start time for each job
  at each work center
• Sequence lists the order in which jobs are to be
  done

9
Terminology Performance Measures

• Average WIP level .(is exactly what it sounds
  like)
• Flowtime The amount of time a job spends from
  the moment it is ready for processing until its
  completion, and includes any waiting time prior
  to processing
• Average WIP level is directly related to the time
  jobs spend in the shop (flowtime)
• Makespan The total time for all jobs to finish
  processing
• For a single machine problem, the makespan is the
  same regardless of the schedule, assuming we do
  not allow any idle time between jobs

10
Terminology Performance Measures

• Performances that have to do with each jobs due
  date
• Lateness The amount of time a job is past its
  due date
• Lateness is a negative number if a job is early
• Earliness The amount of time a job a early
• Tardiness Equals to zero if job is on time or
  early, and equals to lateness if the job is late
• Measures of the cost of production
• Machine utilization and labor utilization are
  primary measures of shop utilization

11
Deterministic Scheduling of a Single Machine
Priority Sequencing Rules

• Random Choose the next job at random. Do not use
  it!
• FCFS First Come First Served. Jobs processed in
  the order they arrive to the shop. Viewed as a
  fair rule.
• SPT Shortest Processing Time. Jobs with the
  shortest processing time are scheduled first.
  Popular method to determine the next homework
  assignment by many students.

12
Deterministic Scheduling of a Single Machine
Priority Sequencing Rules

• SWPT Shortest Weighted Processing Time. A weight
  is assigned to each job based on the jobs value
  (holding cost) or on its cost of delay
• EDD Earliest Due Date. Jobs are sequenced
  according to their due dates.
• CR Critical Ratio. Compute the ratio of
  processing time of the job and remaining time
  until the due date. Schedule the job with the
  largest CR value next, however, if the job is
  late, the ration will be negative, or the
  denominator will be zero, and this job should be
  given highest priority
• (Processing time remaining until completion) /
  (Due Date Current Time)

13
FCFS Example
Flowtime The amount of time a job spends from
the moment it is ready for processing until its
completion, and includes any waiting time prior
to processing Earliness The amount of time a job
a early
Job j pj Dj Cj Fj Lj Ej Tj
1 7 8 7 7 -1 1 0
2 1 12 8 8 -4 4 0
3 5 6 13 13 7 0 7
4 2 4 15 15 11 0 11
5 6 18 21 21 3 0 3
Average 12.8 3.2 1 4.2
Max 21 11 4 11
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SPT Example
Shortest Processing Time

• is optimal for minimizing
• Average and Total flowtime
• Average waiting time
• Average and Total lateness

Job j pj Dj Cj Fj Lj Ej Tj
2 1 12 1 1 -11 11 0
4 2 4 3 3 -1 1 0
3 5 6 8 8 2 0 2
5 6 18 14 14 -4 4 0
1 7 8 21 21 13 0 13
Average 9.4 -0.2 3.2 3
Max 21 13 11 13
15
SWPT Example
Shortest Weighted Processing Time -total weighted
down time -sequencing
Job j Pj Dj wj pj//wj Cj Fj wjFj Lj Ej Tj
4 2 4 5 0.4 2 2 10 -2 2 0
2 1 12 2 0.5 3 3 6 -9 9 0
5 6 18 4 1.5 9 9 36 -9 9 0
3 5 6 3 1.67 14 14 42 8 0 8
1 7 8 2 3.5 21 21 42 13 0 13
Ave 9.8 27.2 0.2 4 4.2
Max 21 42 13 9 13
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EDD Example

• Earliest Due Date

Job j pj Dj Cj Fj Lj Ej Tj
4 2 4 2 2 -2 2 0
3 5 6 7 7 1 0 1
1 7 8 14 14 6 0 6
2 1 12 15 15 3 0 3
5 6 18 21 21 3 0 3
Average 11.8 2.2 0.4 2.6
Max 21 6 2 6
17
CR Example

• Critical Ratio

Subtract Current Time
Job j pj Dj CRj
1 7 8 0.875
2 1 12 0.083
3 5 6 0.833
4 2 4 0.500
5 6 18 0.333
Job j pj Dj Dj-CT CRj
2 1 12 5 0.200
3 5 6 -1 -5.000
4 2 4 -3 -0.667
5 6 18 11 0.545
Schedule jobs 1 ?4 ? 3 ? 2 ? 5
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CR Example (cont)

• Critical Ratio

Job j pj Dj Cj Fj Lj Ej Tj
1 7 8 7 7 -1 1 0
4 2 4 9 9 5 0 5
3 5 6 14 14 8 0 8
2 1 12 15 15 3 0 3
5 6 18 21 21 3 0 3
Average 13.2 3.6 0.2 3.8
Max 21 8 1 8
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Comparing Methods
Method Fj Lj Ej Tj
FCFS Ave 12.8 3.2 1 4.2
Max 21 11 4 11
SPT Ave 9.4 -0.2 3.2 3
Max 21 13 11 13
SWPT Ave 9.8 0.2 4 4.2
Max 21 13 9 13
EDD Ave 11.8 2.2 0.4 2.6
Max 21 6 2 6
CR Ave 13.2 3.6 0.2 3.8
Max 21 8 1 8
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Results for Single Machine Sequencing

• The rule that minimizes the mean flow time of all
  jobs is SPT.
• The following criteria are equivalent
• Mean flow time
• Mean waiting time.
• Mean lateness
• Moores algorithm minimizes number of tardy jobs
• Lawlers algorithm minimizes the maximum flow
  time subject to precedence constraints.

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EDD Example

• Earliest Due Date

Job j pj Dj Cj Fj Lj Ej Tj
4 2 4 2 2 -2 2 0
3 5 6 7 7 1 0 1
1 7 8 14 14 6 0 6
2 1 12 15 15 3 0 3
5 6 18 21 21 3 0 3
Average 11.8 2.2 0.4 2.6
Max 21 6 2 6
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Minimizing the Number of Tardy Jobs

• Morre Algorithm - minimizes number of tardy jobs
• Step 1 Sequence the jobs according to EDD rule
  and initially put all jobs in set V
• Step 2 Find the first tardy job in set V say it
  is job k in the sequence. If there are no
  tardy jobs in the set V, stop the sequence is
  optimal
• Step 3 Select the job with largest processing
  time among first k jobs. Place this job in set U.
  Go to step 2
• Comments
• 1. Placing a job in set U means that it will be
  tardy and will occupy a position in sequence
  after all non-tardy jobs
• 2. Tardy jobs may be schedules in any order
  because the performance measure is the number of
  tardy jobs

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Example Moore Algorithm
Alt. solution 1 4 2 5 3
Job j pj Dj Cj Fj Lj Ej Tj
4 2 4 2 2 -2 2 0
3 5 6 7 7 1 0 1
1 7 8 14 14 6 0 6
2 1 12 15 15 3 0 3
5 6 18 21 21 3 0 3
Average 11.8 2.2 0.4 2.6
Max 21 6 2 6
Iteration 1
Iteration 2
4 2 4 2 2 -2 2 0
1 7 8 9 9 1 0 1
2 1 12 10 10 -2 2 0
5 6 18 16 16 -2 2 0
3 5 6 21 21 15 0 15
Iteration 3
4 2 4 2 2 -2 2 0
2 1 12 3 3 -9 9 0
5 6 18 9 9 -9 9 0
3 5 6 14 14 8 0 8
1 7 8 21 21 13 0 13
Average 9.8 0.2 4 4.2
Max 21 13 9 13
24
Lawlers Algorithm minimizes the maximum flow
time subject to precedence constraints.

• Goal Scheduling a set of simultaneously
  arriving tasks on one machine with precedence
  constraints to minimize maximum lateness
  (tardiness).
• Precedence constraints occur when certain jobs
  must be completed before other jobs can begin.
• Algorithm
• Tasks are scheduled in reverse order job to be
  completed last is scheduled first.
• At each step selection is made from the jobs
  that are not required to precede any other
  unscheduled job.
• Select a job that achieves

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Lawlers Example
Processing for all jobs is 1 day

• One machine ? Ffinal 111111 6
• Select from jobs 4,5,6 such that gives
• min6-3, 6-5, 6-60 ? job 6 is a last job

2) Recalculate F F 6-1 5 Select
from jobs 3,4,5 such that gives min5-4,
5-3, 5-50 ? order x-x-x-x-5-6
3) Recalculate F F 5-1 4 Select
from jobs 3,4 such that gives min4-4,
4-30 ? order x-x-x-3-5-6
4) Recalculate F F 4-1 3 Select
from jobs set 4 ? order x-x-4-3-5-6
5) Recalculate F F 3-1 2 Select
from jobs set 2 ? order x-2-4-3-5-6
6) Recalculate F F 2-1 1 Select
from jobs set 1 ? order 1-2-4-3-5-6
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Lawlers Example
Production is done in next order 1 2 4 3
5 6
Lawlers algorithm minimizes the maximum flow
time subject to precedence constraints
Processing for all jobs is 1 day
Job j pj Dj Cj Fj Lj Ej Tj
1 1 2 1 1 -1 1 0
2 1 5 2 2 -3 3 0
4 1 3 3 3 0 0 0
3 1 4 4 4 0 0 0
5 1 5 5 5 0 0 0
6 1 6 6 6 0 0 0
Average 3.5 -0.67 0.67 0
Max 6 0 3 0
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Gantt Charts

• Pictorial representation of a schedule is called
  Gantt Chart
• The purpose of the chart is to graphically
  display the state of each machine at all times
• Horizontal axis time
• Vertical axis machines 1, 2, , m

Processing Job 1 Job 2
Machine 1 3 5
Machine 2 2 1
Processing Job 1 Job 2
Machine 1 3/1 5/2
Machine 2 2/2 1/1
Question Is it an optimal schedule? Are there
any precedence constrains?
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Gantt Charts
Processing Job 1 Job 2
Machine 1 3 5
Machine 2 2 1
Processing Job 1 Job 2
Machine 1 3/1 5/2
Machine 2 2/2 1/1
Question How to determine THE optimal
solution? What makes scheduling problem more
difficult?
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Example
Processing time / machine number Processing time / machine number Processing time / machine number
Job Operation 1 Operation 2 Operation 3 Release date Due date
1 4/1 3/2 2/3 0 16
2 1/2 4/1 4/3 0 14
3 3/3 2/2 3/1 0 10
4 3/2 3/3 1/1 0 8
Completed by 14 11 13 (late) 10 (late)
Find a solution!
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Deterministic Scheduling with Multiple Machines

• For the case of m machines and n jobs, there are
  n! distinct sequenced on each machine
  (permutations), so (n!)m is the total number of
  possible schedules
• For m 3 and n 4, total number of possible
  schedules is 24313,824
• Assume that each job must be processed in the
  order
• First on machine 1, then machine 2.
• The optimal solution for scheduling n jobs on two
  machines to minimize the total flow time is
  always a permutation schedule
• Assume flow shop in each job operations have to
  be done on both machines
• Permutation schedule is when jobs are done in the
  same order on both machines
• This is the basis for Johnsons algorithm

31
Example
MetalFrame makes 4 different types of metal door
frames. Preparing the hinge upright is a two-step
operation. Natural schedule
Jobs Jobs Jobs Jobs
Machines 1 2 3 4 Total time
1 5 4 3 2 14
2 2 5 2 6 15
Is it optimal?
If idle time for machine 2 is equal to zero,
then we have found an optimal solution
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Deterministic Scheduling with Multiple Machines
Johnsons Rule

• Name Machine 1 A, Machine 2 B,
• then ai processing time for job i on A
• and bi processing time for job i on B
• Johnsons Rule says that job i precedes job j in
  the optimal sequence if
• Algorithm
• Step 1 Record the values of ai and bj in two
  columns
• Step 2 Find the smallest remaining value in two
  columns. If this value in column a, schedule this
  job in the first open position in the sequence
  if this value in column b, schedule this job in
  the last open position in the sequence Cross off
  each job as it is scheduled

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Example (cont)
Jobs Jobs Jobs Jobs
Machines 1 2 3 4 Total time
1 5 4 3 2 14
2 2 5 2 6 15
Johnsons schedule 4 x x x
4 x x 3
job A B
1 5 2
2 4 5
3 3 2
4 2 6
4 x 1 3
4 2 1 3
Natural schedule
Johnsons schedule
Is it optimal?
34
Results for Multiple Machines

• For three machines, a permutation schedule is
  still optimal if we restrict attention to total
  flow time only (not necessarily the case for
  average flow time).
• Under some circumstances, the two machine
  algorithm can be used to solve the three machine
  case
• Label the machines A, B and C
• or
• Redefine Ai Ai Bi and Bi Bi Ci
• When scheduling two jobs on m machines, the
  problem can be solved by graphical means.

35
Sequencing Theory The Two-Job Flow Shop Problem

• Assume that two jobs are to be processed through
  m machines. Each job must be processed by the
  machines in a particular order, but the sequences
  for the two jobs need not be the same
• Graphical procedure developed by Akers (1956)
• Draw a Cartesian coordinate system with the
  processing times corresponding to the first job
  on the horizontal axis and the processing times
  corresponding to the second job on the vertical
  axis (keeping order)
• Block out areas corresponding to each machine at
  the intersection of the intervals marked for that
  machine on the two axes
• Determine a path from the origin to the end of
  the final block that does not intersect any of
  the blocks and that minimizes the vertical
  movement. Movement is allowed only in three
  directions horizontal, vertical, and 45-degree
  diagonal. The path with minimum vertical distance
  corresponds to the optimal solution

36
Example 8.7 (in the book) A regional
manufacturing firm produces a variety of
household products. One is a wooden desk lamp.
Prior to packing, the lamps must be sanded,
lacquered, and polished. Each operation requires
a different machine. There are currently
shipments of two models awaiting processing. The
times required for the three operations for each
of the two shipments are The order of
operations is the same for both jobs A ? B ? C ?
Job 1 Job 1 Job2 Job2
Operation Time Operation Time
Sanding (A) 3 A 2
Lacquering (B) 4 B 5
Polishing (C) 5 C 3
37
Minimizing the flow time is equivalent to finding
the path from the origin to the upper right point
F (for this problem it is art the end of block C)
that maximizes the diagonal movement and
therefore minimizes either the horizontal or the
vertical movement.
Job 1 Job 1 Job2 Job2
Time Time
A 3 A 2
B 4 B 5
C 5 C 3
10 (6)16
12 (3)15
38
Example
142218
14418
Job 1 Job 1 Job2 Job2
Order Operation Time Order Operation Time
B 3 A 2
D 4 D 5
C 2 B 4
A 5 C 3
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Schematic of a Typical Assembly Line

• The problem of balancing an assembly line is a
  classic engineering problem
• A set of n distinct tasks that must be
  completed on each item
• The time required to complete task i is a known
  constant ti
• The goal is to organize the tasks into groups,
  with each group of tasks being performed at a
  single workstation
• The amount of time allotted to each workstation
  is determined in advance
• (C cycle time), based on the desired rate
  of production of the assembly line

40
Assembly Line Balancing

• Assembly line balancing is traditionally thought
  of as a facilities design and layout problem
• There are a variety of factors that contribute to
  the difficulty of the problem
• Precedence constrains some tasks may have to be
  completed in a particular sequence
• Zoning restriction Some tasks cannot be
  performed at the same workstation
• Let t1, t2, , tn be the time required to
  complete the respective tasks

41
Assembly Line Balancing

• The total work content (time) associated with the
  production of an item, say T, is given by

• For a cycle time of C, the minimum number of
  workstations possible is T/C, where the
  brackets indicate that the value of T/C is to be
  rounded to the next larger integer
• Ranked positional weight technique the method
  places a weight on each task based on the total
  time required by all of the succeeding tasks.
  Tasks are assigned sequentially to stations based
  on these weights

42
Assembly Line Balancing

• Example 8.11
• The Final assembly of Noname personal
  computers, a generic mail-order PC clone,
  requires a total of 12 tasks. The assembly is
  done at the Lubbock, Texas, plant using various
  components imported from the Far East. The
  network representation of this particular problem
  is given in the following figure.

43
Assembly Line Balancing

• Precondition
• The job times and precedence relationships for
  this problem are summarized in the table below.

Task Immediate Predecessors Time
1 _ 12
2 1 6
3 2 6
4 2 2
5 2 2
6 2 12
7 3, 4 7
8 7 5
9 5 1
10 9, 6 4
11 8, 10 6
12 11 7
44
Assembly Line Balancing Helgeson and Birnie
Heuristic (1961)

• Ranked positional weight technique
• The solution precedence requires determining the
  positional weight of each task.
• The positional weight of task i is defined as the
  time required
  to perform task i plus the times
  required to perform
  all tasks having
  task i as a predecessor.

Task Time Positional Weight
1 12 70
2 6 58
3 6 31
4 2 27
5 2 20
6 12 29
7 7 25
8 5 18
9 1 18
10 4 17
11 6 13
12 7 7
t3 t7 t8 t11 t12 31
The ranking 1, 2, 3, 6, 4, 7, 5, 8, 9, 10, 11, 12
? ti 70, and the production rate is a unit per
15 minutes The minimum number of workstations
70 / 15 5
45
Assembly Line Balancing Helgeson and Birnie
Heuristic (1961)
Station 1 2 3 4 5 6
Tasks 1 2, 3, 4 5, 6, 9 7, 8 10, 11 12
Processing time 12 14 15 12 10 7
Idle time 3 1 0 3 5 8
Task Immediate Predecessors Time
1 _ 12
2 1 6
3 2 6
4 2 2
5 2 2
6 2 12
7 3, 4 7
8 7 5
9 5 1
10 9, 6 4
11 8, 10 6
12 11 7

• C15

The ranking 1, 2, 3, 6, 4, 7, 5, 8, 9, 10, 11, 12
46
Helgeson and Birnie Heuristic (1961)
C15
Station 1 2 3 4 5 6
Tasks 1 2,3,4 5,6,9 7,8 10,11 12
Processing time 12 14 15 12 10 7
Idle time 3 1 0 3 5 8
Evaluate the balancing results by the efficiency
?ti/NC The efficiencies for C15 is 77.7,
C16 is 87.5, and C13 is 89.7 ?is the best one
Cycle Time15
T26
T112
T26
T36
T42
T52
T52
T612
T91
T85
T77
T104
T104
T116
T127
T127
47
Helgeson and Birnie Heuristic (1961)
C15
Station 1 2 3 4 5 6
Tasks 1 2,3,4 5,6,9 7,8 10,11 12
Processing time 12 14 15 12 10 7
Idle time 3 1 0 3 5 8
C16
Increasing the cycle time from 15 to 16, the
total idle time has been cut down from 20
min/units to 10 ? improvement in balancing
rate. The production rate has to be reduced from
one unit/15 minutes to one unit/16minute
Station 1 2 3 4 5
Tasks 1 2,3,4,5 6,9 7,8,10 11,12
Idle time 4 0 3 0 3
C13
Station 1 2 3 4 5 6
Tasks 1 2,3 6 4,5,7,9 8,10 11,12
Idle time 1 1 1 1 4 0
48
Helgeson and Birnie Heuristic (1961)
C15
Station 1 2 3 4 5 6
Tasks 1 2,3,4 5,6,9 7,8 10,11 12
Processing time 12 14 15 12 10 7
Idle time 3 1 0 3 5 8
C16
13 minutes appear to be the minimum cycle time
with six station balance. Increasing the number
of stations from 5 to 6 results in a great
improvement in production rate
Station 1 2 3 4 5
Tasks 1 2,3,4,5 6,9 7,8,10 11,12
Idle time 4 0 3 0 3
C13
Station 1 2 3 4 5 6
Tasks 1 2,3 6 4,5,7,9 8,10 11,12
Idle time 1 1 1 1 4 0
49
Stochastic Scheduling Static Case

• Single machine case Suppose that processing
  times are random variables. If the objective is
  to minimize average weighted flow time, jobs are
  sequenced according to expected weighted SPT.
  That is, if job times are t1, t2, . . ., and the
  respective weights are u1, u2, . . . then job i
  precedes job i1 if
• E(ti)/ui lt E(ti1)/ui1
• Multiple Machines Requires the assumption that
  the distribution of job times is exponential,
  (memoryless property). Assume parallel processing
  of n jobs on two machines. Then the optimal
  sequence is to to schedule the jobs according to
  LEPT (longest expected processing time first).
• Johnsons algorithm for scheduling n jobs on two
  machines in the deterministic case has a natural
  extension to the stochastic case as long as the
  job times are exponentially distributed.

50
Stochastic Scheduling Queueing Theory

• A typical queueing process
• The basic phenomenon of queueing arises whenever
  a shared facility needs to be accessed for
  service by a large number of jobs or customers.
  (Bose)
• The study of the waiting times, lengths, and
  other properties of queues. (Mathworld)
• Applications
• Telecommunications Health services
• Traffic control Predicting computer
  performance
• Airport traffic, airline ticket sales
  Layout of manufacturing systems
• Determining the sequence of computer operations

51
Examples of Queueing Theory
http//www.bsbpa.umkc.edu/classes/ashley/Chaptr14/
sld006.htm
52
Stochastic Scheduling Dynamic Analysis

• View network as collections of queues
• FIFO data-structures
• Queuing theory provides probabilistic analysis of
  these queues
• Typical operating characteristics of interest
  include
• Lq Average number of units in line waiting for
  service
• L Average number of units in the system (in
  line waiting for service and being serviced)
• Wq Average time a unit spends in line waiting
  for service
• W Average time a unit spends in the system
• Pw Probability that an arriving unit has to
  wait for service
• Pn Probability of having exactly n units in the
  system
• P0 Probability of having no units in the system
  (idle time)
• U Utilization factor, of time that all
  servers are busy

53
Characteristics of Queueing Processes

• Arrival pattern of customers
• Service pattern of servers
• Queue discipline
• System capacity
• Number of service channels
• Number of service stages

54
Characteristics of Queueing Processes

• Arrival pattern of customers
• Probability distribution describing the times
  between successive customer arrivals
• Time independent ?Stationary arrival patterns
• Time dependent ? Non-stationary
• Batch or Bulk customer arrivals
• Probability distribution describing the size of
  the batch
• Customers behavior while waiting
• Wait no matter how long the queue becomes
• If the queue is too long, customer may choose not
  to enter into the system
• Enter, wait, and choose to leave without being
  serviced
• If there is more than one waiting line, customer
  may switch jockey

55
Characteristics of Queueing Processes

• Arrival pattern of customers
• Service pattern of servers
• Single or Batch
• May depend on the number of customers waiting ?
  state dependent
• Stationary or Non-stationary
• Queue discipline
• Manner in with customers are selected to service
• First Come First Served (FCFS)
• Last Come First Served (LCLS)
• Random Selection for Service (RSS)
• Priority Schemes
• Preemptive case
• Non-preemptive case

56
Characteristics of Queueing Processes

• Arrival pattern of customers
• Service pattern of servers
• Queue discipline
• System capacity
• Finite queueing situations Limiting amount of
  waiting room
• Number of service channels
• Single-channel system
• Multi-channel system, generally assumed that
  parallel channels operate independently of
  each other
• Number of service stages

57
Notation Used in Queueing Processes

• Full notation A / B / X / Y / Z
  Shorthand A / B / X
• A indicates the interarrival-time
  distribution Assumes Y is infinity,
• B the probability distribution for service
  time Z FCFS
• X number of parallel service channels
• Y the restriction on system capacity
• Z the queue discipline (FCFS)

Symbol Explanation
A B M Exponential, D Deterministic, Ek Erlang type Hk Mixture of k exponentials, PH Phase type, G General
X Y 1, 2, ... , infinity 1, 2, ... , infinity
Z FCFS, LCLS, RSS, PR priority, GD general discipline
58
Queueing Processes Littles Formulas

• One of the most powerful relationships in
  queueing theory was developed by John D.C. Little
  in the early 1960s.
• Formulas
• and ,
• where ? is an average rate of customers entering
  the system, and
• W is an expected time customer will spend in
  the system

59
Poisson Process Exponential Distribution

• M stands for "Markovian", implying exponential
  distribution for service times or inter-arrival
  times, that carries the memoryless property
• past state of the system does not help to predict
  next arrival / departure

60
Calculating Expected System Measures for M/M/1

• The utilization rate ? ? / µ
• P0 1 ?
• Pi ?i(1 ?), for i 1, 2, 3,
• these formulas hold only if lltm

CHARACTERISTIC SYMBOL FORMULA Utilization
? ? / µ Exp. No. in System L
? / (µ ?) ? / (1-?) Exp. No. in Queue
Lq ?2/ µ(µ ?) ?2 / (1-?) Exp. Waiting
Time WL/ ? 1 / (µ ?) ? / ?(1-?) Exp.
Time in Queue WqLq/ ? ? / µ(µ ?) ?2 /
?(1-?) Prob. System is Empty P0 1 (? / µ)
1 - ?
61
Calculating Expected System Measures for M/M/m
http//www.ece.msstate.edu/hu/courses/spring03/no
tes/note4.ppt
62
Calculating Expected System Measures for M/M/m

• Assumption
• - m servers
• - all servers have the same service rate µ
• - single queue for access to the servers
• - arrival rate ?n ?
• - departure rate

63
Calculating Expected System Measures for M/M/m

64
Example

• Unisex hair salon runs on a first-come,
  first-served basis. Customers seem to arrive
  according to a Poisson process with mean arrival
  rate of 5/hr. Because of Ms. H.R. Cutts
  excellent reputation, customers are always
  willing to wait. Average service time of 10 min
  is exponentially distributed.
• Calculate the average number of customers in the
  shop and the average number of customers waiting
  for a haircut.
• Calculate the percentage of time an arrival can
  walk right in without having to wait at all.
• The waiting room has only 4 seats. What is the
  probability that a customer upon arrival rill
  have to stand?
• Calculate average system waiting time, and the
  line delay.

65
Other Systems
M/M/1/K - system with a capacity K ?eff
effective arrival rate M/D/1 M/G/1
M/G/8 Assignment download the QTS add-in for
Excel software to check the homework problems
answers http//www.geocities.com/qtsplus/Download
Instructions.htmDOWNLOAD_INSTRUCTIONS
66
Homework Assignment

• Read Ch. 8 (8.1 8.10)
• Read Supplement Two (S2.1 - S2.13)
• 8.4, 8.5, 8.7, 8.12, 8.15,
• 8.18, 8.23, 8.25, 8. 27, 8.28

67
References

• Presentation by McGraw-Hill/Irwin
• Presentation by Professor JIANG Zhibin,
  Department of Industrial Engineering
  Management, Shanghai Jiao Tong University
• Production Operations Analysis by S.Nahmias
• Production Planning, Control, and Integration
  by Sipper and Bulfin Jr.
• Inventory Management and Production Planning and
  Scheduling by Silver, Pyke and Peterson
• Fundamentals of Queueing Theory by Cross and
  Harris
• http//www.geocities.com/qtsplus/DownloadInstructi
  ons.htmDOWNLOAD_INSTRUCTIONS QTS analysis for
  Excel