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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 management of waiting lines.
Waiting Lines - Queuing TheoryConcept of loss of business due to customers waiting Cost analysis of provision of faster servicing to reduce queue
length Marginal cost of extra provisioning during rush hours
The Purpose of Queuing Models
Queuing models are used to:describe the behavior of queuing systems determine the level of service to provide evaluate alternate configurations for providing service
Queuing System Cost
Cost of providing the service also known as service cost
Cost of not providing the service also known as waiting cost
Trade-offCost of operating service facility
Total Expected Cost
Cost of providing service Cost of waiting time
Optimal Service Level
Important factors of Queuing Situations Arrival
Service pattern Queue discipline Customers behavior Maximum number of customers allowed in the system Nature of Calling Source
Queuing System: General Structure Arrival Process According to source According to numbers According to time
Service System Single server facility Multiple, parallel facilities with single queue Multiple, parallel facilities with multiple queues Service facilities in a parallel
Common Queuing System ConfigurationsCustomer Arrives Customer Arrives ...Waiting Line Server
Customer Leaves Customer Leaves Customer Leaves Customer Leaves Customer Leaves Customer Leaves Customer Leaves Customer Leaves
...Waiting Line Server 1 Server 2
...Waiting Line Server 2 Server 3
... Customer ArrivesWaiting Line Server 1 Server 2 Server 3
Queue Structure First come first served Last come first served Service in random order Priority service
Customer Behavior Balking Reneging Jockeying Collusion
Characteristics of Queuing Systems: The Arrival Process Arrival rate - the manner in which customers arrive at the system for service. Arrivals are often described by a Poisson random variable: X
e P ( Xcustomers ) = for X = 0,1, 2, L X!
wherex = no. of arrival per unit(e.g. hour) P(x) = probability of exactly x arrivals =average arrival rate (e.g., calls arrive at a rate of =5 per hour) e = 2.7183 (known as exponential constatnt)
Characteristics of Queuing Systems: The Arrival Process
Arrivals are often described by a Poisson random X variable: T
P( X ) =
T for X = 0,1, 2, L X!
where T = period e.g 30 minutes period
Characteristics of Queuing Systems: The Service Process Service time - the amount of time a customer spends receiving service (not including time in the queue). Service times are often described by an Exponential random variable: P(service time more than t) = et, for t 0 P(service time less than t) = 1-et, for t 0 Exponential probability distribution used in describing service times.
where is the service rate (e.g., calls can be serviced at a rate of =7 per hour) The average service time is 1/.
Comments If arrivals follow a Poisson distribution with mean , inter arrival times follow an Exponential distribution with mean 1/. Example Assume calls arrive according to a Poisson distribution with mean =5 per hour. Inter arrivals follow an exponential distribution with mean 1/5 = 0.2 per hour. On average, calls arrive every 0.2 hours or every 12 minutes. The exponential distribution exhibits the Markovian (memory less) property.
Problem 1 On an average 5 customers reach a barbers shop every hour. Determine the probability that exactly 2 customers will reach in a 30 minutes period.
Problem 2The manager of a bank observes that, on average,18 customers are served by a cashier in an hour. Assuming that the service time has an exponential distribution, what is the probability that (a) a customer will be free within 3 minutes,(b)a customer shall be served in more than 12 minutes.
Kendall Notation Queuing systems are described by 3 parameters: A/B/s Parameter AM = Markovian interarrival times D = Deterministic interarrival time
Parameter BM = Markovian service times G = General service times D = Deterministic service times
Parameter sA number Indicating the number of servers.
Operating CharacteristicsTypical operating characteristics of interest include: P0 Lq L - Utilization factor, % of time that all servers are busy. - Prob. that there are no zero units in the system. - Avg number of units in line waiting for service. - Avg number of units in the system (in line & being served). Wq - Avg time a unit spends in line waiting for service. W - Avg time a unit spends in the system (in line & being served). Pw - Prob. that an arriving unit has to wait for service. Pn - Prob. of n units in the system.
Model 1: Poisson-exponential single server model infinite populationAssumptions:
Arrivals are Poisson with a mean arrival rate of, say Service time is exponential, rate being Source population is infinite Customer service on first come first served basis Single service station For the system to be workable,
Model 2: Poisson-exponential single server model finite populationHas same assumptions as model 1, except that population is finite
Model 3: Poisson-exponential multiple server model infinite populationAssumptions
Arrival of customers follows Poisson law, mean rate Service time has exponential distribution, mean service rate There are K service stations A single waiting line is formed Source population is infinite Service on a first-come-first-served basis Arrival rate is smaller than combined service rate of all service facilities
Model: 1 Operating Characteristics
a) Queue lengthaverage number of customers in queue waiting to get service
b) System lengthaverage number of customers in the system
c) Waiting time in queueaverage waiting time of a customer to get service
d) Total time in systemaverage time a customer spends in the system
e) Server idle timerelative frequency with which system is idle
= mean number of arrivals per time period (eg. Per hour) = mean number of customers served per time period
Probability of system being busy/traffic intensity = /
Probability of an empty facility/system being idle P(0) = 1 / =1- Probability that exactly one customer in the system P(1) = P(0) Probability that exactly two customer in the system P(2) = P(1)= 2P(0) Probability that n customer in the system P(n) = nP(0)
Probability of having exactly n customers in the system P(n) = nP(0)=n (1- ) Expected no. of customers in the system Ls= This can be solve to obtain
Expected number of customers in the system Ls = / (- )
Expected number of customers in the queue(including empty queue) = Expected no. of customer in the systemexpected no. of customers being served i.e. Lq= Ls- / i.e. Lq= / (- ) - / i.e. Lq = 2/ (- )= 2/ 1- Expected number of customers in the non empty i.e. Lq = / - = 1/ 1-
Average waiting time in queue = product of expected queue length and expected time between arrivals Wq= 1/ . Lq Wq= 1/ . 2/ (- ) Wq= / (- ) Average waiting time in the system = product of expected no. of customers in the system and expected time between arrivals Ws= 1/ . Ls Ws= 1/ . / (- ) Ws= 1/ -
Probability that a customer spends more than t units of time in the system = Ws(t)= et/Ws Probability that a customer spends more than t units of time in the queue = Wq(t)= et/Ws
Problem 3. A
television repairman finds that the time spent on his job has an exponential distribution with a mean of 30 minutes .If he repairs sets in the order in which they came in,and if arrival of sets follows a Poisson distribution with an average rate of 10 per day ,what is the repairmen's expected idle time each day? How many jobs are ahead of the average set just brought in? Assuming he works for 8 hours a day.
Problem 3. A
television repairman finds that the time spent on his job has an exponential distribution with a mean of 30 minutes .If he repairs sets in the order in which they came in,and if arrival of sets follows a Poisson distribution with an average rate of 10 per hour day ,what is the repairmen's expected idle time each day? How many jobs are ahead of the average set just brought in? Assuming he works for 8 hours a day.
Problem 3. A tailor specializes in ladies dresses. The number of customers approaching the Tailor appear to be Poisson distributed with a mean of 6 customers per hour. The tailor attends the customer on a first come first serve basis. The tailor can attend the customers at an average rate of 10 customers per hour with a service time exponentially distributed.Required 1. Find the probability of no. of arrivals(0 through 5) (i) a 15 minutes interval (ii) a 30 minutes interval. 2. (i)The utilization parameter. (ii)probability that system remains idle. 3. Average time that the tailor is free on a 10 hour working day 4. probability of no. of arrivals(0 through 5) in the system 5. What is the expected no. of customers (i)in the tailor sho