 1  Outline  simulating GI/G/1 queues  M/M/1 queues  theoretical results of queueing systems  an inventory system  simulation program with an event calendar  hand simulation

 2  To Simulate an GI/G/1 Queue  definition of variables definition of variables  definition of events definition of events  pseudo code pseudo code  flow chart of program flow chart of program NKL TaTa TsTs TNOW TMAX distribution of inter-arrival times distribution of service times

 3  Flow Chart for Simulating an GI/G/1 Queue N = L = K = T = 0; T a = random inter- arrival time; T s = infinity Yes ouput L/T now and L/K L = L + (T s - T now )N; T now = T s ; N = N - 1 L = L + (T a - T now )N; T now = T a ; N = N + 1; K = K +1; T a = T now + random inter-arrival time T s = T now + random service time T now  T max No next event service N = 0 T s = T now + random service time No Yes T s = infinity arrival N = 1 No

 4  To Simulate an GI/G/1 Queue  definition of variables  definition of events  pseudo code  flow chart of program  Java program Java program

 5  Variations of GI/G/1 Queues  M/M/1-based queue  standard: Java programJava program  with queue limit: Java programJava program

 6  Terms Associated with a Customer  for customer i  arrival time, t ai  time to start service, t si  time to leave system, t di  time in queue, D i = t si – t ai  service time, S i = t di – t si  time in system, W i = t di – t ai other names of time in queue = waiting time in queue = delay time other names of time in system = waiting time in system = total time in system

 7  Machine (Server) Queue (FIFO) Theoretical Result: Conservation of Time n n-1n-2... nn-1... n n n-1 n-2n-3... for each customer, his time in system = time in queue + service time ave. time in system = ave. time in queue + E(service time) tag nth customer

 8  Sample-Path Results of a Queueing System  state: N(t), number of customers in system N(t)N(t) t average # of customers up to t = 26:

 9  Sample-Path Results of a Queueing System  total length of rectangular boxes of the same color = waiting time of the corresponding customer in system average # in system = (arrival rate) (average waiting time)

 10  Theoretical Result: Little ’ s Formula  very general result  for any system: L = w  L = average # of customers in system  = arrival rate of customers  w = average time in system

 11  Theoretical Result: Little ’ s Formula  GI/G/1 queue  the whole system  average # in system = (arrival rate) (average time in system)  the waiting queue as the system  average # in queue = (arrival rate) (average time in queue)  the server as the system  utilization of server = (arrival rate) E(service time)

 12  An Inventory System Day n demand of the Day n D n ~ Geo(0.1) at the beginning of Day n: update inventory if ordered quantity arrives lost sales, no back order $0.1 per unit on hand at the beginning of a day after receiving any outstanding order (Q, s) policy: order Q items at the end of a day if inventory on hand  s at most one outstanding order leadtime of orders: equally likely to be 0 to 4 days (i.e., the ordered quantity can possibly arrive the next day) ordering cost: $20 per order penalty: $1 per piece lost

 13  Inventory System  inventory system inventory system  definition of variables definition of variables  definition of events definition of events  pseudo code pseudo code  flow chart flow chart  Java program Java program

 14  Simulating Complex Systems  simple system (up to now)  chess piece, machine, GI/G/1 queue, inventory system  tailor-made programs for these problems  complex problems  generic program structure  event scheduled by an event calendar

 15  Generic Program Structure for Complex Systems  Initialization Event while (not stopping (e.g., TNOW < TTERM) ) { switch (Next Event Type) { case 1: tasks of type 1 event; break;.... case n: tasks of type n event; break; } } Termination Event need to keep track of the timing of events

 16  Conceptual Structure of an Event Calendar  event: (time, type, tasks to do)  event calendar: a link list with scheduled future events in ascending order of time  example  four events scheduled at epoch 0  event 1: (24.3, type = 4, tasks to do)  event 2: (35.6, type = 1, tasks to do)  event 3: (41.3, type = 2, tasks to do)  event 4: (5000, type = END, tasks to do = end program)

 17  Event Calendar of a Simulation Program Event. Cal. TNOW = 0: type 4 tasks to do type 1 tasks to do type 2 tasks to do END End simulation

 18  35.6type 1 tasks to do 41.3type 2 tasks to do 5000END End simulation 35.6type 1 tasks to do 41.3type 2 tasks to do 5000END End simulation 24.3type 4 tasks to do 35.6type 1 tasks to do 41.3type 2 tasks to do 5000END End simulation Event Calendar of a Simulation Program Event. Cal. move time to TNOW = type 1 tasks to do 41.3type 2 tasks to do 5000END End simulation suppose that no new event is created by a type 4 event execute first event and update event calendar 35.6type 1 tasks to do 41.3type 2 tasks to do 5000END End simulation TNOW = 0:

 19  24.3type 4 tasks to do 35.6type 1 tasks to do 41.3type 2 tasks to do 5000END End simulation Event Calendar of a Simulation Program Event. Cal. move time to TNOW = suppose that a new event is created by an type 4 event at 39.1 execute first event and update event calendar TNOW = 0: 35.6type 1 tasks to do 41.3type 2 tasks to do 5000END End simulation 39.1type 4 tasks to do 35.6type 1 tasks to do 41.3type 2 tasks to do 5000END End simulation 39.1type 4 tasks to do 35.6type 1 tasks to do 41.3type 2 tasks to do 5000END End simulation 39.1type 4 tasks to do

 20  Operations of the Event Calendar  delete an event that has (just) occurred  after completing all tasks at TNOW, remove the first event in the event calendar  insert a new event  get the event time of the new event  check where to insert the event  move backward one rank for events occurring after the new event  insert the new event Describe concept only; actual implementation depends on the data structure used

 21  A Crude Pseudo-Code of GI/G/1 Queue with Event Calendar  1  initialization  set TNOW = 0; initialize variables; set termination event; set initial events  2  get next event type  3  while { next event != termination } {  get next event time and next event details  set TNOW to next event time  update the Event Calendar  execute next event, including updating variables and possibly adding new events to Event Calendar  get next event type }}  4  execute the termination Event

 22  An GI/G/1 Queue with the Event Calendar  inter-arrival times ~ exp of mean 10  service times ~ exp of mean 5  Java Program Java Program

 23  Simulation by Hand  simulate the system for 20 time units to get  total # of customers served (in the time horizon)  average and maximum waiting time in queue  time-average and maximum number in queue  average and maximum total time in system  utilization (proportion of busy time of station)

 24  Input Data  Initially (time 0) empty and idle  Part NumberArrival TimeInterarrival TimeService Time  Stop when 20 minutes of (simulated) time have passed

 25  hand simulation