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  1. 1. QUEUING THEORY [M/M/C MODEL] Student Adviser:-Assist.Prof. Sanjay Kumar Student:-Ram Niwas Meena Semester:-Fourth“Delay is the enemy of efficiency” and “Waiting is the enemy of utilization”
  2. 2. OVERVIEW What is queuing theory? Examples of Real World Queuing Systems? Components of a Basic Queuing Process A Commonly Seen Queuing Model Terminology and Notation Little’s Formula The M/M/1 – model Example M/M/c Model 1
  3. 3. WHAT IS QUEUING THEORY? Mathematical analysis of queues and waiting times in stochastic systems.  Used extensively to analyze production and service processes exhibiting random variability in market demand (arrival times) and service times. Queues arise when the short term demand for service exceeds the capacity  Most often caused by random variation in service times and the times between customer arrivals.  If long term demand for service > capacity the queue will explode! Queuing theory is the mathematical study of waiting lines (or queues) that enables mathematical analysis of several related processes, including arriving at the (back of the) queue, waiting in the queue, and being served by the Service Channels at the front of the queue. 2
  4. 4. What do you mean by Balking, Reneging, Jockeying?BalkingIf a customer decides not to enter the queue since it is too long is calledBalkingRenegingIf a customer enters the queue but after sometimes loses patience and leaves itis called RenegingJockeyingWhen there are 2 or more parallel queues and the customers move from onequeue to another is called JockeyingWhat is Transient & Steady State of the system? Queuing analysis involves the system’s behavior over time. If the operatingcharacteristics vary with time then it is said to be transient state of thesystem. the behavior becomes independent of its initial conditions (no. of Ifcustomers in the system) and of the elapsed time is called Steady Statecondition of the system 3
  5. 5. QUEUING MODELS CALCULATE: Average number of customers in the system waiting and being served Average number of customers waiting in the line Average time a customer spends in the system waiting and being served Average time a customer spends waiting in the waiting line or queue. Probability no customers in the system Probability n customers in the system Utilization rate: The proportion of time the system is in use. 4
  6. 6. Examples of Real World Queuing Systems? Commercial Queuing Systems  Commercial organizations serving external customers  Ex. , bank, ATM, gas stations… Transportation service systems  Vehicles are customers or servers  Ex. Vehicles waiting at toll stations and traffic lights, trucks or ships waiting to be loaded, taxi cabs, fire engines, elevators, buses … Business-internal service systems  Customers receiving service are internal to the organization providing the service  Ex. Inspection stations, conveyor belts, computer support … Social service systems  Ex. Judicial process, hospital, waiting lists for organ transplants or student dorm rooms … 5
  7. 7. Arrivals Population of from the Queue Service dirty cars (waiting line) facility Exit the system general population … Prabhakar Car Wash enter exit Arrivals to the system In the system Exit the systemArrival Characteristics Waiting Line Service Characteristics•Size of the population Characteristics •Service design•Behavior of arrivals •Limited vs. unlimited •Statistical distribution of•Statistical distribution •Queue discipline serviceof arrivals
  8. 8. Components of a Basic Queuing Process (II) The calling population  The population from which customers/jobs originate  The size can be finite or infinite (the latter is most common)  Can be homogeneous (only one type of customers/ jobs) or heterogeneous (several different kinds of customers/jobs) The Arrival Process  Determines how, when and where customer/jobs arrive to the system  Important characteristic is the customers’/jobs’ inter-arrival times  To correctly specify the arrival process requires data collection of inter arrival times and statistical analysis. 7
  9. 9. Components of a Basic Queuing Process (III) The queue configuration  Specifies the number of queues  Single or multiple lines to a number of service stations  Their location  Their effect on customer behavior  Balking and reneging  Their maximum size (# of jobs the queue can hold)  Distinction between infinite and finite capacity 8
  10. 10. Components of a Basic Queuing Process (IV) The Service Mechanism  Can involve one or several service facilities with one or several parallel service channels (servers) - Specification is required  The service provided by a server is characterized by its service time  Specification is required and typically involves data gathering and statistical analysis.  Most analytical queuing models are based on the assumption of exponentially distributed service times, with some generalizations. The queue discipline  Specifies the order by which jobs in the queue are being served.  Most commonly used principle is FIFO.  Other rules are, for example, LIFO, SPT, EDD…  Can entail prioritization based on customer type. 9
  11. 11. A Commonly Seen Queuing Model (I) The Queuing System The Service Facility C S = Server C S The QueueCustomers (C) • CCC…C • • C S Customer =C 11
  12. 12. A Commonly Seen Queuing Model (II) Service times as well as inter arrival times are assumed independent and identically distributed  If not otherwise specified Commonly used notation principle: (a/b/c):(d/e/f)  a = The inter arrival time distribution  b = The service time distribution  c = The number of parallel servers  d= Queue discipline  e = maximum number (finite/infinite) allowed in the system  f = size of the calling source(finite/infinite) Commonly used distributions  M = Markovian (exponential/possion) –arrivals or departurs distribution Memoryless  D = Deterministic distribution  G = General distribution Example: M/M/c  Queuing system with exponentially distributed service and inter- arrival times and c servers 12
  13. 13. Example – Service Utilization Factor• Consider an M/M/1 queue with arrival rate = and service intensity =• = Expected capacity demand per time unit• = Expected capacity per time unit Capacity Demand λ ρ Available Capacity μ• Similarly if there are c servers in parallel, i.e., an M/M/c system but the expected capacity per time unit is then c* Capacity Demand Available Capacity c* 14
  14. 14. Terminology and Notation The state of the system = the number of customers in the system Queue length = (The state of the system) – (number of customers being served)n=Number of customers/jobs in the system at time tPn(t) =The probability that at time t, there are n customers/jobs in the system. n =Average arrival intensity (= # arrivals per time unit) at n customers/jobs in the system n =Average service intensity for the system when there are n customers/jobs in it. =The utilization factor for the service facility. (= The expected fraction of the time that the service facility is being used) 13
  15. 15. Notation For Steady State Analysis Pn = The probability that there are exactly n customers/jobs in the system (in steady state, i.e., when t ) L = Expected number of customers in the system (in steady state) Lq = Expected number of customers in the queue (in steady state) W = Expected time a customer spends in the system Wq= Expected time a customer spends in the queue 15
  16. 16. Little’s Formula Assume that n = and n= for all n L W Lq Wq Assume that n is dependent on n Let Pn n n 0 L W Lq Wq 16
  17. 17. The M/M/1 - modelAssumptions - the Basic Queuing Process Infinite Calling Populations  Independence between arrivals The arrival process is Poisson with an expected arrival rate  Independent of the number of customers currently in the system The queue configuration is a single queue with possibly infinite length  No reneging or balking The queue discipline is FIFO The service mechanism consists of a single server with exponentially distributed service times  = expected service rate when the server is busy 17
  18. 18. The M/M/1 Model n= and n = for all values of n=0, 1, 2, … 0 1 2  n- n n+1 1 Steady State condition: =( / )<1 P0 = 1- Pn = n(1- ) P(n k) = k L= /(1- ) Lq= 2/(1- ) = L- W=L/ =1/( - ) Wq=Lq/ = /( ( - )) 18
  19. 19. Example – SMS Hospital Situation  Patients arrive according to a Poisson process with intensity ( the time between arrivals is exp( ) distributed.  The service time (the doctor’s examination and treatment time of a patient) follows an exponential distribution with mean 1/ (=exp( ) distributed) The SMS can be modeled as an M/M/c system where c=the number of doctors Data gathering = 2 patients per hour = 3 patients per hour Questions – Should the capacity be increased from 1 to 2 doctors? – How are the characteristics of the system ( , Wq, W, Lq and L) affected by an increase in service capacity? 19
  20. 20. Summary of Results – SMS Hospital Interpretation  To be in the queue = to be in the waiting room  To be in the system = to be in the ER (waiting or under treatment) Characteristic One doctor (c=1) Two Doctors (c=2) 2/3 1/3 P0 1/3 1/2 (1-P0) 2/3 1/2 P1 2/9 1/3 Lq 4/3 patients 1/12 patients L 2 patients 3/4 patients Wq 2/3 h = 40 minutes 1/24 h = 2.5 minutes W 1h 3/8 h = 22.5 minutes Is it warranted to hire a second doctor ? 20
  21. 21. Generalized Poisson queuing model In steady state the following balance equation must hold for every state n (proved via differential equations) The Rate In = Rate Out Principle: Mean entrance rate = Mean departure rate• In addition the probability of being in one of the states must equal 1 Pi 1 i 0 21
  22. 22. Generalized Poisson queuing model State Balance Equation 0 0 1 P1 0 P0 P1 P0 1 1 0 P0 2 P2 1 P1 1 P1 P2 1 P1 2   n n 1 Pn 1 n 1 Pn 1 ( n n ) Pn Pn n 1 Pn 1   n 0 0 1 0 1 2 Normalizat ion : Pi P0 1  1 i 0 1 1 2 1 2 3 22 C0 C2
  23. 23. Steady State Measures of Performance Steady State Probabilities n P0 1 Pn P0 Expected Number of customers in the System and in the Queue  Assuming c parallel servers Lq (n c ) Pi L n Pn n c n 0 23
  24. 24. COMPONENTS OF A QUEUING SYSTEM Service Process Servers Queue or Waiting LineArrival Process Exit
  25. 25. The M/M/c Model (I)• Generalization of the M/M/1 model – Allows for c identical servers working independently from each other 0 1 2  c-2 c-1 c c+1 2 (c-2) (c-1) c c 1 c 1 n c ( / ) ( / ) 1 P0 n 0 n! c! 1 ( /( c ) Steady State Condition: n ( / ) =( /c )<1 P0 for n 1, 2 ,  , c n! Pn n ( / ) 25 P0 for n c 1, c 2 , n c c! c
  26. 26. The M/M/c Model (II)• A Condition for existence of a steady state solution is that = /(c ) <1 c ( / ) Lq (n c ) Pn ... P0 2 n c c ! (1 ) Little’s Formula Wq=Lq/ W=Wq+(1/ ) Little’s Formula L= W= (Wq+1/ ) = Lq+ / 26
  27. 27. The M/M/c/K – Model (I) An M/M/c model with a maximum of K customers/jobs allowed in the system  If the system is full when a job arrives it is denied entrance to the system and the queue. Interpretations  A waiting room with limited capacity (for example, the ER at County Hospital), a telephone queue or switchboard of restricted size  Customers that arrive when there is more than K clients/jobs in the system choose another alternative because the queue is too long (Balking) 27
  28. 28. The M/M/c/K – Model (II) The state diagram has exactly K states provided that c<K 0 1 2  c-1 c  K-1 K 2 3 (c-1) c c c The general expressions for the steady state probabilities, waiting times, queue lengths etc. are obtained through the balance equations as before (Rate In = Rate Out; for every state) 28
  29. 29. The M/M/c/ /N – Model (I) An M/M/c model with limited calling population, i.e., N clients A common application: Machine maintenance  c service technicians is responsible for keeping N service stations (machines) running, that is, to repair them as soon as they break  Customer/job arrivals = machine breakdowns  Note, the maximum number of clients in the system = N Assume that (N-n) machines are operating and the time until breakdown for each machine i, Ti, is exponentially distributed (Ti exp( )). If U = the time until the next breakdown U = Min{T1, T2, …, TN-n} U exp((N-n) )). 29
  30. 30. The M/M/c/ /N – Model (II)• The State Diagram (c service technicians and N machines) – = Arrival intensity per operating machine – = The service intensity for a service technician N (N-1) (N-(c-1)) 0 1 2  c-1 c  N-1 N 2 3 (c-1) c c• General expressions for this queuing model can be obtained from the balance equations as before 30
  31. 31. Reference:-1. Operations research: an introduction By:- Hamdy A. Taha2 . Operations research By:-P.Sankar Iyar