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Lecture 9

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MKTG 2030
Ben Kelly

Omis  Lecture  9                 Jessica  Gahtan     Waiting  Lines/Queuing  Theory  (Ch.  13.0 -­‐13.6,  13.9-­‐13.12)   Introduction  to  Queuing  Theory   • It  is  estimated  that  Americans  spend  a  total  of  37  billion  hours  a  year  waiting  in  lines.   • Places  we  wait  in  line...   -­‐  Stores       -­‐  hotels       -­‐  post  offices   -­‐  Banks       -­‐  traffic  lights   -­‐  restaurants   -­‐  Airports     -­‐  theme  parks   -­‐  on  the  phone   • Waiting  lines  do  not  always  contain  people...   -­‐ Returned  videos     -­‐ Subassemblies  in  a  manufacturing  plant   -­‐ Electronic  message  on  the  Internet   • Queuing  theory  deals  with  the  analysis  and  managem ent  of  waiting  lines.      Queuing  theory  useful  in  both  manufacturing     and  service  areas    Queuing  models  are  used  to:     – Describe  the  behavior  of  queuing  systems   – Determine  the  level  of  service  to  provide   – Evaluate  alternate  configurations  for  providing  service   Lines  in  Operations  Management       Assembly  lines       Production  lines       Trucks  waiting  to  unload  or  load       Workers  waiting  for  parts       Customers  waiting  for  products       Broken  equipment  waiting  to  be  fixed       Customers  waiting  for  service   Common  Queuing  Situations     Server-­‐  does  transaction   Customers  –  those  things  that  are  lining  up   Characteristics  of  Waiting-­‐Line  Systems   1. Arrivals  or  inputs  to  the  system   Page  1   Omis  Lecture  9                 Jessica  Gahtan      Population  size,  behavior,  statistical  distribution  –  if  a  population  is  large  enough  can   treat  as  being  infinite  –  if  it’s  a  small  population  –  if  10/20  are  there,  chances  are  smaller   that  someone  else  might  show  up   2. Queue  discipline,  or  the  waiting  line  itself    Limited  or  unlimited  in  length,  discipline  of  people  or  items  in  it  –  if  limited  length  –  only   certain  number  of  things  can  come  in   –  i.e.  busy  signal  on  a  phone  if  no  free  lines   3. The  service  facility    Design,  statistical  distribution  of  service  times   Parts  of  a  Waiting  Line  –  anything  served  or  in  queue  (not  yet  served)  =  system     Arrival  Characteristics   1. Size  of  the  population    Unlimited  (infinite)  or  limited  (finite)   Example  of  Unlimited  Population:   Shoppers  arriving  at  a  supermarket   Example  of  Limited  Population:   5  fax  machines     -­‐>  Each  fax  machine  is  a  potential  customer  (it  may  break  down  and  require  servic e)   2. Pattern  of  arrivals    Scheduled      Random  (the  occurrence  can  not  be  predicted)   • Arrival  rate  -­‐  the  manner  in  which  customers  arrive  at  the  system  for  service.     Exponential  Distribution   -­‐ t Density:  f(t)  =  λe   Mean:     1/  λ   2     Variance:   1/  λ where  λ  is  the  arrival  rate   • Time  between  arrivals  is  often  modeled  using  an  exponential  distribution   • Why?       Exponential  distribution  has  a  memoryless  property   -­‐  the  probability  distribution  on  the  time   until  the  next  arrival  does  not  depend  on  how  long  it  has  been  since  the  la st  arrival,  which   significantly  simplifies  the  analysis  of  a  waiting  line  system   Page  2   Omis  Lecture  9                 Jessica  Gahtan     Exponential  and  Poisson   • If  interarrival  times  are  exponentially  distributed  with  parameter   λ,  then  number  of  arrivals  per   unit  time  follows  a  Poisson  distribution  with  paramet er  λt.   Poisson  Distribution   • The  number  of  arrivals  per  unit  of  time  can  be  estimated  by  a  distribution  known  as  the   Poisson   Distribution.         Example   What  is  the  probability  that  5  students  arrive  in  any  random  hour  to  register  in  a  course  when  the  averag e   arrival  rate  is  3  students  per  hour  and  interarrival  times  are  exponentially  distributed?       P(x)   =   ?   x   =   number  of  arrivals  per  unit  of  time  =  5   λ   =   Average  arrival  rate  =  3   e       =   0.0498       3. Behaviour  of  Arrivals   • Most  queuing  models  assume  that  an  arriv ing  customer  is  a  patient  customer.  (Waits  in  the   queue  and  does  not  switch  lines)   • Life  is  complicated  by  the  fact  that  people  have  been  known  to  balk  or  to  renege.   Page  3   Omis  Lecture  9               Jessica  Gahtan      Balk  –  refuse  to  join  a  waiting  line  because  it  is  too  long  to  suit  their  needs.  –  Reduces   amount  you  serve  (when  line  gets  to  a  certain  length)   –  dissatisfied  cust  -­‐  ignoring    Renege  –  enter  the  queue  but  then  become  inpatient   and  leave  the  line.     Waiting  Line  Characteristics    Limited  or  unlimited  queue  length    Small/limited  number  of  waiting  chai rs  in  a   doctor  clinic.    Even  though  many  queues  may  have  practical  limits,  if  the  limit  is  large  we  can  treat  the   queue  as  unlimited.    Queue  discipline  (order  of  service)     FIFO  (FCFS):  First  in,  first  out.  (First  come,  first  served).  This  is  the  most  common   assumption.     LIFO  (Last-­‐in,  First-­‐out)     RANDOM:  Select  arrival  to  serve  at  random  from  those  waiting.    PRIORITY:  Give  some  arrivals  priority  for  service.   Service  Characteristics   Service  Times   Service  time  -­‐  the  amount  of  time  a  customer  spends  receiving  ser vice  (not  including  time  in  the  queue).      Service  time  distribution    Constant  service  time  -­‐  Non-­‐random      Automatic  car  wash      Industrial  robots  engaged     in  vehicle  assembly      Random:  Use  Negative  Exponential  probability  distribution.    Mean  service  rate  =  m    e.g:  4  customers/hr    Mean  service  time  =  1/m    e.g:  1/4  hour         =  15  minutes    Continuous  distribution:     Page  4   Omis  Lecture  9                 Jessica  Gahtan         Negative  Exponential  Distribution        Queuing  system  designs   Channels  -­‐  How  many  paths  are  there  through  the  system  –  AFTER  you  are  in  line?     Phases  -­‐  How  many  stops  must  a  customer  make?    (Single  phase  means  only  one  stop  for  service.)     • Single-­‐channel  system,  multiple-­‐channel  system   • Single-­‐phase  system,  multiphase  system   Page  5   Omis  Lecture  9                 Jessica  Gahtan     Single-­‐channel,  Single-­‐phase         Single-­‐channel,  Multiphase  phase  system     Page  6   Omis  Lecture  9                 Jessica  Gahtan     Multi-­‐channel,  single-­‐phase  system         Multi-­‐channel,  multiphase  system       Common  Queuing  System  Configurations   Page  7   Omis  Lecture  9                 Jessica  Gahtan       Why  the  other  line  is  likely  to  move  faster ?       Performance  of  Queuing  Systems   Measuring  Queue  Performance   Queuing  analysis  can  obtain  many  measures  of  a  waiting -­‐line  system’s  performance.   1. Average  time  that  each  customer  or  object  spends  in  the  queue   2. Average  queue  length   3. Average  time  each  customer  spends  in  the  system   4. Average  number  of  customers  in  the  system   5. Probability  that  the  service  facility  will  be  idle   6. Utilization  factor  for  the  system   7. Probability  of  a  specific  number  of  customers  in  the  system   Operating  Characteristics   Typical  operating  characteristics  of  interest  include:   ρ  =U    Utilization  factor,  %  of  time  that  all  servers  are  busy.   P 0    e  percentage  of  the  time  there  are  no  customers/units  in  the  system  (Probability  of  idle  service   facility)   L -­‐  Avg  number  of  units  in  line  waiting  for  service.   q     L=L s   -­‐  Avg  number  of  units  in  the  system  (in  line  &  being  served).   W q     -­‐  Avg  time  a  unit  spends  in  line  waiting  for  service.   W=W s   -­‐  Avg  time  a  unit  spends  in  the  system    (in  line  &  being  served).   P w     -­‐  Prob.  that  an  arriving  unit  has  to  wait  for  service.   Page  8   Omis  Lecture  9                 Jessica  Gahtan     P n     -­‐  Prob.  of  n  units  in  the  system.   P   -­‐  Probability  of  more  than  k  units  in  system  (Also,  fraction   of  time  there  are  more  than  k  units  in   n >  k       the  system)     Queuing  Costs   Operations  managers  must  recognize  the  trade -­‐off  that  takes  place  between  two  costs:     1. The  cost  of  providing  good  service     2. The  cost  of  customer  or  machine  waiting  time.     -­‐ Managers  want  queues  that  are  short  enough  so  that  customers  do  not  become  unhappy  and   either  leave  without  buying  or  buy  but  never  return.     -­‐ However,  managers  may  be  willing  to  allow  some  waiti ng  if  it  is  balanced  by  a  significant  savings   service  costs.       Kendall  Notation   • Queuing  systems  are  described  by  3  parameters:  1/2/3     – Parameter  1         M  =  Markovian  interarrival  times  ( Negative  exponential  distribution                        (Poisson  arrivals))         D  =  Deterministic  interarrival  times   – Parameter  2         M  =  Markovian  service  times         G  =  General  service  times         D  =  Deterministic  (scheduled)  service  times   – Parameter  3:  A  number  Indicating  the  number  of  servers.  Labeled  as  S     Examples:   M/M/3       D/G/4       M/G/2   Page  9   Omis  Lecture  9                 Jessica  Gahtan     Types  of  Queuing  Models   A  wide  variety  of  queuing  models  may  be  applied  in  operations  management.  We  will  focus  on  the   following  models:      Simple  (M/M/1)    [(M/M/S  )  when  #  of  servers=1]    Example:  Information  booth  at  the  mall    Multi-­‐channel  (M/M/S)  [when  S>1]    Example:  Airline  ticket  counter    General  Service  (M/G/1)    Example:  Car  oil  change    Constant  Service  (M/D/1)    Example:  Automated  car  wash       Assumptions  in  the  Basic  Model  M/M/1    Customer  population  is  homogeneous  and  infinite    Queue  capacity  is  infinite    Customers  are  well  behaved  (no  balking  or  reneging)    Arrivals  are  served  FIFO  (FCFS)    Poisson  arrivals.    The  time  between  arrivals  follows  a  negative  exponential  distribution    Exponential  service  times:  Services  are  described  by  the  negative  exponential  distribution     Steady  State  Assumptions    Mean  arrival  rate   λ,  mean  service  rate  μ,     and  the  number  of  servers  are  constant.    The  service  rate  is  greater  than  the  arrival  rate.           You  should  always  check  this.           Consider  –  what  would  happen  if  this  were  not  the  case?    These  conditions  have  existed  for  a  long  time.   Simple  (M/M/1)  Model  Characteristics   • Type:  Single-­‐channel,  single-­‐phase  system   • Input  source:  Infinite;  no  balks,  no  reneging   • Arrival  distribution:  Poisson   • Queue:  Unlimited;  single  line   • Queue  discipline:  FIFO     • Service  distribution:  Negative  exponential   Page  10   Omis  Lecture  9                 Jessica  Gahtan       Model  A  –  Single-­‐Channel  (M/M/1)                         Remember:        λ  &  µ  Are  Rates    λ=  Mean  number  of  arrivals  per  time  period       e.g.,  3  units/hour   μ  =  Mean  number  of  people  or  items  served  per  time  period     e.g.,  4  units/hour     1/  μ  =  15  minutes/unit   Page  11   Omis  Lecture  9                 Jessica  Gahtan     Single  Channel  In  Class  Example  1   The  manager  of  a  video  store  is  interested  in  providing   good  service.       On  a  Friday  or  Saturday  night,  on  average  30  customers  per  hour  arrive  at  the  counter  to  check  out  a   video.    The  customers  are  served  at  an  average  rate  of  35  customers  per  hour  from  a  single  cash  register.     Time  between  arrivals  follows  an  exponential  distribution,  as  does  service  time.   Determine  the  operating  characteristics  for  the  video  store.   1. Average  number  of  customers  in  line   2. Average  number  of  customers  in  the  system   3. Average  wait  time  in  line   4. Average  time  in  the  system   5. Server  utilization   6. Probability  of  0  customers  are  in  the  system     7. Probability  that  more  than  6  customers  are  in  the  system   8. Probability  that  there  are  exactly  2  customers  in  the  system     The  Q.xls  Queuing  Template   • Formulas  for  the  operating  characteristics  of  a  number  of  qu euing  models  have  been  derived   analytically.   • An  Excel  template  called  Q.xls  implements  the  formulas  for  several  common  types  of  models.   • Q.xlsx  was  created  by  Professor  David  Ashley  of  the  Univ.  of  Missouri  at  Kansas  City.   In  Class  Example  1  –  Q.xls     Single  Channel  Example   The  Case:  customers  arriving  at  a  repair  shop  with  1  service  bay   Page  12   Omis  Lecture  9                 Jessica  Gahtan                
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