1. QUEUING THEORY

2. Queuing Theory
Each of us has spent a great deal of time waiting in lines.
In this chapter, we develop mathematical models for waiting lines, or queues.
To describe a queuing system, an input process and an output process must be specified.
Examples of input and output processes are:
Situation
Input Process
Output Process
Bank
Customers arrive at bank
Tellers serve the customers
Pizza hut
Request for pizza delivery are received
Pizza parlor send out scooters to deliver pizzas
QUEUING THEORY

3. Queuing Costs vs. Level of Service
Total expected cost
Cost of operation
Cost of operating the service facility per unit time
Cost of waiting customers per unit time
Optimum service level
Level of Service
QUEUING THEORY

4. QUEUING SYSTEM Input source Queuing Process Service Process Jockey
Service System
Waiting Customers
Queuing discipline
Arrival Process
Departure
Input source
Queuing Process
Service Process
Jockey
Balk
Renege
QUEUING THEORY

5. Input Source Characteristics
Input source is characterized by size of calling population, behavior of the arrivals and pattern of arrival of customers.
Size of calling population – Homogeneous, sub-classes, finite or infinite.
Behavior of arrivals – Patient customer, balking, reneging or jockeying.
Pattern of arrivals – Static or dynamic, pattern is further subdivided based on nature of arrival rate and control on the arrival.
QUEUING THEORY

6. Pattern of Arrival Process
In static systems the arrivals can be random or at constant rate.
Random arrivals can be varying with time.
To analyze the queuing system we should know the probability distribution of the arrivals.
We obtain distribution for the arrivals and also for the inter-arrival time.
In dynamic system, both service facility and customers are controlled.
The arrival pattern can be approximated by Poisson, Erlang or Exponential distribution.
QUEUING THEORY

7. QUEUING PROCESS
Queuing process refers to length of queues and number of queues.
There can be single server (facility) or multiple facility.
Queues can be finite (cinema hall) or infinite (sales orders).
When customers experience long queues then they may not enter the queue.
Queue discipline controls the service for the customers
QUEUING THEORY

8. QUEUE DISCIPLINE
The queue discipline describes the method used to determine the order in which customers are served.
The most common queue discipline is the FCFS discipline (first come, first served), in which customers are served in the order of their arrival.
Under the LCFS discipline (last come, first served), the most recent arrivals are the first to enter service.
If the next customer to enter service is randomly chosen from those customers waiting for service it is referred to as the SIRO discipline (service in random order).
QUEUING THEORY

9. QUEUEING DISCIPLINE Finally we consider priority queuing disciplines.
A priority discipline classifies each arrival into one of several categories.
Each category is then given a priority level, and within each priority level, customers enter service on an FCFS basis.
Another factor that has an important effect on the behavior of a queuing system is the method that customers use to determine which line to join.
QUEUING THEORY

10. SERVICE PROCESS Service process is characterized by following
arrangement of service facility.
distribution of service times.
server’s behavior
management policies.
Arrangement of facilities can be
single queue and single server
single queue and multiple server
mixed arrangement
QUEUING THEORY

11. Service Process Distribution
Time taken be server from the commencement to completion of service for a customer is called service time.
To describe the output process of a queuing system, we usually specify a probability distribution – the service time distribution – which governs a customer’s service time
The most commonly distribution used for service time distribution is negative exponential.
Servers are in parallel if all servers provide the same type of service and a customer needs only pass through one server to complete service.
Servers are in series if a customer must pass through several servers before completing service.
QUEUING THEORY

12. Performance measures of Queuing theory
Time related questions
What is the average time an arriving customer has to wait in the queue (Wq) before being served.
What is the average time an arriving customer spends in the system (Ws) including waiting and service.
Quantitative questions related to no of customers
Expected no of customers in the queue for service (Lq)
Expected no of customers who are in the system waiting in queue or being serviced (Ls)
QUEUING THEORY

13. Performance measures of Queuing theory
Questions involving time both for customers and servers
Probability that an arriving customer has to wait before being served Pw.
Probability that a server is busy at any particular point (). It is the proportion of the time that a server spends actually with a customer.
Probability of having ‘n’ customers in the system when it is in steady state. Pn, n= 0,1,…n.
Probability of customer denied service because of queue being full. Pd.
QUEUING THEORY

14. Poisson Distribution
Customers arrive at a bank or grocery in a completely random fashion.
The probability density function for describing such arrivals during a specified period follows Poisson distribution.
Let x be the arrivals that take place during a specified time units.
Poisson pdf is given by
Mean and variance is given by
QUEUING THEORY

15. Exponential Distribution
If the no of arrivals at a service facility during a specified period occurs according to Poisson distribution then the distribution of the intervals between successive arrivals must follow the negative exponential distribution.
If  is the rate at which Poisson events occur then the distribution of the time ,x, between successive arrivals is given as
QUEUING THEORY

16. Analysis of a Simple Queue
Arrivals with an average arrival rate of 
service rate  at the server
Single server
Infinite no of waiting customers.
QUEUING THEORY

17. Steady State & Transient State
At the start of the system the queue is influenced by no of customers, and the elapsed time.
This is referred as transient state.
After sufficient time the system returns to steady state.
We analyse the steady state behavior as transient state is complex and beyond our scope.
Arrival rate has to be less than service rate else the system cannot reach steady state.
QUEUING THEORY

18. Modeling Arrival Process
We assume that the arrival process can be approximated by Poisson distribution.
Let us consider a Poisson process involving the number of arrivals n over a time period t.
If  is the arrival rate (traffic intensity) then the no of customers expected in time t will be t.
The probability mass function Pn is given by
QUEUING THEORY

19. Modeling Arrival Process
The probability mass function of no arrival (n=0) in time t is given by
Let us define the random variable T as the time between successive arrivals.
T will be a continuous random variable.
The assumption that each inter-arrival time is governed by the same random variable implies that the distribution of arrivals is independent of the time of day or the day of the week.
This is the assumption of stationary inter-arrival times.
QUEUING THEORY

20. Modeling Arrival Process
Stationary inter-arrival times is often unrealistic, but we may often approximate reality by breaking the time of day into segments.
A negative inter-arrival time is impossible. This allows us to write
We define1/λ to be the mean or average inter-arrival time.
QUEUING THEORY

21. Modeling Arrival Process
The distribution of the random variable T is called the exponential distribution.
Its probability density function is given by f(t) = λe-λt. for t 0 and is o for t<0.
We can show that the average or mean inter-arrival time is given by
QUEUING THEORY

22. Derivation of Exp Distribution
The exponential distribution is based on three axioms:
Given N(t), the number of events during the interval (0,t), the probability process describing N(t) has stationary independent increments, in the sense that the probability of an event occurring in interval (T, T+S) depends only on the length of S.
The probability of an event occurring in a sufficiently small time interval h> 0 is positive but less than 1
In a sufficiently small time interval h>0, at most one event can occur-that is P{N(h)>1}=0
QUEUING THEORY

23. Derivation of Exponential Distribution
Define Pn(t) as the probability of n events occurring during t.
We have seen earlier that probability of no arrival in time t is given by P0(t) = e -t
Define f(t) as the pdf of the interval t between successive events, t>0 . We then have
P(inter-event time >T) = P(no event during T)
QUEUING THEORY

24. No Memory Property of Exp Distribution
If T has an exponential distribution, then for all nonnegative values of t and h,
A density function that satisfies the equation is said to have the no-memory property.
The no-memory property of the exponential distribution is important because it implies that if we want to know the probability distribution of the time until the next arrival, then it does not matter how long it has been since the last arrival.
QUEUING THEORY

25. Markov Process Markov Processes
X(t) satisfies the Markov Property (memoryless) which states that -for any choice of time instants ti, i=1,……, n where tj>tk for j>k
P{X(tn+1)=xn+1| X(tn)=xn ……. X(t1)=x1} =P{X(tn+1)=xn+1| X(tn)=xn}
Memoryless property as the state of the system at future time tn+1 is decided by the system state at the current time tn and does not depend on the state at earlier time instants t1,…….., tn-1
QUEUING THEORY

26. Modeling the Service Process
We assume that the service times of different customers are independent random variables and that each customer’s service time is governed by a random variable S having a density function s(t).
We let 1/µ be the mean service time for a customer.
The variable 1/µ will have units of hours per customer, so µ has units of customers per hour. For this reason, we call µ the service rate.
Unfortunately, actual service times may not be consistent with the no-memory property.
In certain situations, inter-arrival or service times may be modeled as having zero variance; in this case, inter-arrival or service times are considered to be deterministic.
QUEUING THEORY

27. Modeling the Service Process
For example, if inter-arrival times are deterministic, then each inter-arrival time will be exactly 1/λ, and if service times are deterministic, each customer’s service time is exactly 1/µ.
Negative exponential distribution is used to describe the service times distribution.
The pdf of the service times is
The probability of n service completion in time t is given by
QUEUING THEORY

28. Basic Notations
QUEUING THEORY

29. Relationship among Performance measures
We have the following relationships
Average no of customers in the system is equal to the expected no of customers in queue plus in service
Average waiting time of the customer in the system is equal to waiting time in queue plus in service
QUEUING THEORY

30. Relationship among Performance measures
Average no of customers served per busy period is given by
Average length of queue during busy period is given by
QUEUING THEORY

31. Relationship among Performance measures
Little's law gives a very important relation between Ls, the mean number of customers in the system, Ws, the mean sojourn time and , the average number of customers entering the system per unit time. Little's law states that
The probability , Pn of n customers in the queuing system at any time can be used to determine all the basic measures of performance in the following order
QUEUING THEORY

32. Basic Axioms
Let Pn(t) denote the probability that there are n customers in the system at time t.
The rate of change of Pn(t) with respect to time t is denoted by derivative of Pn(t) with respect to t.
In case of steady state we have
QUEUING THEORY

33. Basic Axioms
The no of arrivals in non-overlapping intervals are statistically independent.
The probability that two or more customers arrive in time interval (t, t+t) is negligible and is denoted by 0 t. 
The probability that a customer arrives in the time interval (t, t+t) is P1(t) =  t + 0 t.
 is inter-arrival time and is independent of total no of arrivals up to time t. As t  0, the qty 0 t is negligible.
QUEUING THEORY

34. Probability Analysis Assume that, as t 0
P{one arrival in time  t} =   t
P{no arrival in time  t} =1-   t
P{more than one arrival in time  t} = O(( t)2) = 0
P{one departure in time  t} =   t
P{no departure in time  t} =1-   t
P{more than one departure in time  t} = O(( t)2) = 0
P{one or more arrival and one or more departure in time  t} = O(( t)2) = 0
Arrival Process, Mean Inter-arrival time = 1/ 
Service Process, Mean Service time = 1/ 
QUEUING THEORY

35. Waiting Time Paradox
Suppose the time between the arrival of buses at the student center is exponentially distributed with a mean of 60 minutes.
If we arrive at the student center at a randomly chosen instant, what is the average amount of time that we will have to wait for a bus?
The no-memory property of the exponential distribution implies that no matter how long it has been since the last bus arrived, we would still expect to wait an average of 60 minutes until the next bus arrived.
QUEUING THEORY

36. Birth Death Process
We subsequently use birth-death processes to answer questions about several different types of queuing systems.
We define the number of people present in any queuing system at time t to be the state of the queuing systems at time t.
We call Pn the steady state, or equilibrium probability, of state n.
The behavior of Pij(t) before the steady state is reached is called the transient behavior of the queuing system.
A birth-death process is a continuous-time stochastic process for which the system’s state at any time is a nonnegative integer.
QUEUING THEORY

37. Birth Death Process Law 1 Law 2 Law 3
With probability λnΔt+o(Δt), a birth occurs between time t and time t+Δt. A birth increases the system state by 1, to n+1. The variable λn is called the birth rate in state n. In most queuing systems, a birth is simply an arrival.
Law 2
With probability µnΔt+o(Δt), a death occurs between time t and time t + Δt. A death decreases the system state by 1, to n-1. The variable µn is the death rate in state n. In most queuing systems, a death is a service completion. Note that µ0 = 0 must hold, or a negative state could occur.
Law 3
Births and deaths are independent of each other.
QUEUING THEORY

38. Steady State Birth Death Process
We now show how the Pn’s may be determined for an arbitrary birth-death process.
The key role is to relate (for small Δt) Pij(t+Δt) to Pij(t).
The above equations are often called the flow balance equations, or conservation of flow equations, for a birth-death process.
QUEUING THEORY

39. Flow Balancing Approach
In the “rate diagram” given below, think of the following:
Each circle representing a state (i.e., number of customer in the system) has an unknown probability Pn, n= 0, 1, 2, … associated with it 1 2 3   4
QUEUING THEORY

40. Flow Balancing Approach
We obtain the flow balance equations for a birth-death process:
In general we can write
Value of P0 is determined from the equation
QUEUING THEORY

41. The Kendall-Lee Notation for Queuing Systems
Standard notation used to describe many queuing systems.
The notation is used to describe a queuing system in which all arrivals wait in a single line until one of s identical parallel servers is free. Then the first customer in line enters service, and so on.
To describe such a queuing system, Kendall devised the following notation.
Each queuing system is described by six characters: 1/2/3/4/5/6
QUEUING THEORY

42. The Kendall-Lee Notation for Queuing Systems
The first characteristic specifies the nature of the arrival process. The following standard abbreviations are used:
M = Inter-arrival times are independent, identically distributed (iid) and exponentially distributed
D = Inter-arrival times are iid and deterministic
Ek = Inter-arrival times are iid Erlangs with shape parameter k.
GI = Inter-arrival times are iid and governed by some general distribution
QUEUING THEORY

43. The Kendall-Lee Notation for Queuing Systems
The second characteristic specifies the nature of the service times:
M = Service times are iid and exponentially distributed
D = Service times are iid and deterministic
Ek = Service times are iid Erlangs with shape parameter k
G = Service times are iid and governed by some general distribution
The third characteristic is the number of parallel servers.
QUEUING THEORY

44. The Kendall-Lee Notation for Queuing Systems
The fourth characteristic describes the queue discipline:
FCFS = First come, first served
LCFS = Last come, first served
SIRO = Service in random order
GD = General queue discipline
The fifth characteristic specifies the maximum allowable number of customers in the system.
The sixth characteristic gives the size of the population from which customers are drawn.
M/E2/8/FCFS/10/∞ might represent a health clinic with 8 doctors, exponential interarrival times, two-phase Erlang service times, a FCFS queue discipline, and a total capacity of 10 patients.
QUEUING THEORY

45. (M/M/1):(FCFS// ) The models has following assumptions
Poisson arrival rate and exponential distribution of inter-arrival times.
Single waiting line with no restriction on queue length
Queuing discipline is First come first served
Single server with exponential service time.
The solution is obtained by using flow balancing approach. The three states possible in small time Δt before t are
The system is in state n and no arrivals or service completions occurred.
The system is in state (n+1) and a service completion occurs, reducing the customers to n.
The system is in state (n-1) and an arrival occurs, bringing the no of customers to n.
QUEUING THEORY

46. Single Server (M/M/1) Obtaining system of equations
Taking the limits as t 0, and subject to the same normalisation, we get
QUEUING THEORY

47. Single Server Model For equilibrium conditions we require
Let =/. Then we get following results
Applying normalisation condition of Pn=1 we get
QUEUING THEORY

48. Performance Measures – Single Server
Average no of customers in the system
Average no of customers in queue
QUEUING THEORY

49. Performance Measures – Single Server
Average time a customer spends waiting in the queue
Average time a customer spends in system
Probability of customers in system greater than or equal to k
QUEUING THEORY

50. Performance Measures – Single Server
Variance (fluctuations of queue length)
Probability that a queue is non-empty
Expected length of non-empty queue
QUEUING THEORY

51. QUEUING THEORY

52. Example (M/M/1)
At a cycle repair shop on an average a customer arrives every five minutes and on an average, the service time is 4 minutes per customer. Suppose that the inter-arrival time follows Poisson distribution and service time is exponentially distributed. Find the various performance characteristics
QUEUING THEORY

53. M/M/1:N/FCFS/ Limited no of customers are allowed in the system.
Service rate need not be greater than arrival rate.
Steady state equations are
QUEUING THEORY

54. Performance characteristics
Service rate meed not be more than arrival rate as arrivals are controlled.
eff, rather than ,, is the rate that matters.
eff is calculated as arrival rate minus the probability of losing customers.
eff =  - lost =  - Pn= (1-Pn)
Other parameters are calculated from the eff.
QUEUING THEORY

55. Performance Characteristics
QUEUING THEORY

56. Example
Consider a single server queuing system with Poisson arrival, exponential input, exponential service times. Suppose the mean arrival rate is 3 calling units per hour, the expected service time is 0.25 hour and the maximum permissible calling units in the system is 2. Derive the steady state probability distribution of the number of calling units in the system, and then calculate the expected no in the system.
QUEUING THEORY

57. M/G/1: GD/ /
Queuing models in which the arrival and departure rate do not follow Poisson are complex.
The above is a model with Poisson arrival rate and service time t, is represented by any general distribution.
The mean of the distribution is E(t) and variance is Var(t).
Let  be the arrival rate. If E(t) < 1, then we can calculate the Length of the system queue.
QUEUING THEORY

58. Example
QUEUING THEORY