UNIT –I: QUEUEING THEORY
1. Defi Define ne a queu queuee and a cus custo tome mer. r. Queue: The flow of customers waiting for service in in a system rendering some service is called a queue. Customer: The arriving unit requires some service to be performed is ca lled a customer. 2. What are are the basic characteri characteristics stics of a queuing queuing system? system? (i) The in input or arrival pa pattern (ii) The service mechanism (iii) The que queu ue di discipline (iv) The cu customer’s be behavio vior 3. Define Define (i) Arrival Arrival patter pattern n (ii) (ii) Service Service mechanism mechanism (i) (i) Arri Arriva vall patt patter ern: n: Thi Thiss is the the inpu inputt patt patter ern n whic which h desc descri ribe bess the the manne mannerr in which the customers arrive and join the queuing system since the arrival of a customer is always random; the pattern is described in terms of probability distribution. (ii) (ii) Servic Servicee mech mechani anism: sm: The arrange arrangemen mentt made made for servic servicee by the busine business ss firm firm can be represented by means of probability distribution for the number of customers serviced per unit of time or the inter-service time. 4. Defi Define ne queu queuee disc discip ipli line ne This describes the mode in which the customers select the service when the queue has been formed. The most common disciplines are (i) First come first first served (ii) Last come first served (iii) Service in random order. 5. Define Define the followin following g (i) Balkin Balking g (ii) Renegin Reneging g (iii) (iii) Prioritie Prioritiess (iv) Jockeying Jockeying (i) (i) Balk Balkin ing: g: A cus custo tome merr lea leave vess the the queue queue beca becaus usee the the queu queuee is is too too long long and and the the customer has no time to wait or has no sufficient space for waiting. (ii) (ii) Renegi Reneging: ng: This This happens happens when when a custom customer er waiti waiting ng for servic servicee leaves leaves the queue due to impatience. (iii) (iii) Priori Prioritie ties: s: In few situat situation ions, s, some some custo customer merss are are serve served d befor beforee other otherss regardless of their arrival. These customers have priority over others. (iv) (iv) Jockeyi Jockeying: ng: Custom Customers ers may swit switch ch over over from from one wait waiting ing line line to to anoth another. er. This happens in a railway reservation counter. 6. What What is limi limited ted and and unlimi unlimited ted queue queue lengt length? h? Limited queue is a situation where the waiting time is restricted to some maximum length and the unlimited queue is a situation where there are no restrictions on the maximum length of the waiting time.
7. Define Define transi transient ent and steady steady state state A queuing system is said to be in transient state when its operating characteristics are dependent of time otherwise it is steady state.
8. Explain Kendall’s notation for representing queuing models. A queuing model is specified and represented symbolically in the form (a/b/c): (d/e) ; where
a – inter arrival time , b – service mechanism ,
c – number of service, d – the capacity of the system, e – queue discipline. 9. Define traffic intensity or utilization factor. A simple queue can be measured by its traffic intensity is given by
ρ = Mean arrival rate / Mean service rate = λ/µ 10. What are the classifications of queuing models? The queuing models are classified as follows. ( M/M/I) = (∝/FIFO) (i) ( M/M/S) = (∝/FIFO) (ii) (iii) ( M/M/I) = (N/FIFO) (iv) ( M/M/S) = (N/FIFO) Where M represents Markov process indicating the number of arrivals and the completed service in the time follow poison process. The letter I & S represents a single server and multilevel respectively. The fourth letter ∝ and N represent the capacity of the system ie infinite and finite respectively. 11. Give the formula for probability of n units in the system under single server, FCFS discipline. ρ = (1- ρ ) ρn where ρ = λ/µ
λ = mean arrival rate ; µ = mean service rate. 12. Write the formula for finding the (i) Expected number in the queue (ii) Expected number in the non-empty queue (i) Lq = ρ2 / 1-ρ where ρ = λ/µ (ii) Ln = µ / µ - λ 13. Write Little’s formula (i) Ls = λWs (ii) Lq = Ls -
λ/µ = λWq (iii) Ws = ρWq
14. Write the formula to fine the probability of waiting time more than or equal to ‘t’ in the queue
∝ P(waiting time ≥t) = ∫
ρ (µ - λ ) ρ - (µ - λ ) wdw
PART-B 1. Arrivals at the telephone booth are considered to be Poisson with an average time of 12min between one arrival and the next. The length of a phone call is assumed to be distributed exponentially with mean 4min. (a) Find the average number of persons waiting in the system. (b) What is the probability that a person arriving at the booth will have to wait in the queue? (c) Estimate the fraction of the day when the will be in use. (d) What is the average length of the queue? (e) What is the average waiting time of the person in system and queue?
2. Customers arrive at a one-man barber shop according to a Poisson process with a mean inter arrival time of 12min. Customers spend an average of 10min in the barber’s chair. (a) What is the expected number of customers in the barber shop and in the queue? (b) Calculate the percentage of time an arrival can walk straight into the barber’s chair without having to wait. (c) How much time can a customer expect to spend in the barber’s shop? (d) What is the average time customer spends in the queue? (e) Calculate the percentage of customers who have to wait prior to getting into barber’s chair.
3. Customers arrive at a watch repair shop according to a Poisson distribution at a rate of one per every 10 minutes and the service time is an exponential random variable with mean 8 minutes. Find Ls, Lq, Ws, Wq.
4. In a telephone booth the arrivals are on the average 15 per hour. A call on the average takes 3 minutes. If there are just one phone. Find Ls, Lq, Ws, Wq and idle time of the booth.
5. Customers arrive at a one man barber shop according to a Poisson process with an mean inter arrival time of 12 minutes. Customers spend a average of 10 minutes in the barber’s chair. Find Ls, Lq, Ws, Wq.
6. In a case manufacturing plant, a loading crane takes exactly 10 minutes to load a car into a wagon and again comes back to the position to load another car. If the arrival of cars is a Poisson stream at an average rate of one after every 20 minutes, calculate the average time of a car in a stationary state.
7. Arrivals at a telephone booth are considered to be Poisson at the rate of 6 per hour. The length of phone call is assumed to be distributed exponentially with mean of three minutes. (a) What is the probability that a person arriving at the booth will have to wait? (b) The telephone department will install a second booth when convinced that an arrival would expect waiting for at least 3 minutes for a phone call. By how much should the flow of arrivals increase in order to justify a second booth? (c) What is the average length of the queue? (d) What is the probability that it will take more than 10 minutes altogether to wait for the phone and complete the call?
8. Cars arrive at a petrol pump, having one petrol unit, in Poisson fashion with an average of 10 cars per hour. The service time is distributed exponentially with mean of 3 minutes. Find average number of cars in the system, average waiting time in the queue, average queue length, the probability that the number of cars in the system is 2.
9. A supermarket has more than one server servicing at counters. The customer arrive in a Poisson fashion at the rate of 10 per hour. The service time for each customer is expected with mean 4 minutes. Find the probability of a customer has to wait for the service, average queue length, the average time spent by a customer in the queue.
10. A telephone exchange has two long distance operators. The telephone company finds that during the peak load, long distance calls arrive in a Poisson fashion at an average rate of 15 per hour. The length of service on these calls is approximately expo nentially distributed with mean length 5 minutes. (a) What is the probability that a subscriber will have to wait for his long distance call during the peak hours of the day? (b) If the subscribers wait and are serviced in turn, what is the expected waiting time?
11. There are 3 typists in an office. Each typist can type an average of 6 letters per hour. If letters arrive for being typed at the rate of 15 letters per hour. (a) What is the fraction of time all the typists will be busy? (b) What is the average number of letters waiting to be typed? (c) What is the average time a letter has to spend for waiting and for being typed?
12. A general insurance company has three claim adjusters in its branch office. People with claims against the company are found to arrive in Poisson fashion at an average rate of 20 per 8 hour day. The amount of time that an adjuster spends with claimant is found to have negative exponential distribution with mean service time 40 minutes. Claimants are processed in the order of their appearance. How many hours a week can an adjuster5 expect to spend with claimants. How much time, on the average does claimant spend in the branch office.
13. Given an average arrival rate of 20 per hour, is it better for a customer to get service at a single channel with mean service rate of 22 customers per hour or at one of two channels in parallel with mean service rate of 11 customers per hour for each of the two channels. Assume both queues to be of Poisson type.
14. A bank has 2 tellers working on savings accounts. The first teller handles withdrawals only. The second teller handles deposits only. It has been found that the service time distributions for both deposits and withdrawals are exponentially with mean service time of 3 minutes per customer. Depositors are found to arrive in a Poisson fashion throughout the day with mean arrival rate of 16 per hour. Withdrawers also arrive in a Poisson fashion with arrival rate of 14 per hour. (a) What would be the effect of the average waiting time for the customers if each teller could handle both withdrawals and deposits? (b) What would be the effect, if this could only be accomplished by increasing the service time to 3.5 Minutes?
15. Trains arrive at the year every 15 minutes and the service time is 33 minutes. If the line capacity at the yard is limited to 4 trains, find (a) The probability that the yard is empty. (b) The average number of trains in the system.
16. A car park contains 5 cars. The arrival of cars is Poisson at a mean rate of 10 per hour. The length of time each car spends in the car park is exponentially distributed with mean of 5 hour. How many cars are there in the car park on an average? 17. In a railway marshalling yard, goods trains arrive at the rate of 30 trains per day. Assume that the inter arrival time follows as exponential distribution and the service time is also to be assumed as exponential with mean of 36 minutes. Calculate (a) the probability that the yard is empty. (b) The average queue length, assuming the line capacity of the year is 9 trains. 18. Patients arrive at a clinic according to Poisson distribution at a rate of 30 patients per hour. The waiting room does not accommodate more than 14 patients. Examination time per patient is exponential with mean rate of 20 per hour. (a) Find the effective arrival rate at the clinic. (b) What is the probability that an arriving will not wait? (c) What is the expected waiting time until a patient is discharged from the clinic?
19. In a single server queuing system with Poisson input and exponential service times, if the mean arrival rate is 3 calling units per hour, the expected service time is 0.25 hour and the maximum possible number of calling units in the system is 2, find Pn and Average number of calling units in the system. 20. A 2-person barbershop has 5 chair to accommodate waiting customers. Potential customers, who arrive when all 5 chairs are full, leave without entering barbershop.
Customers arrive at the average rate of 4 per hour and spend an average of 12 min in the barber’s chair. Compute P0, P1, P7, Lq , Ls, Ws and Wq.
21. A car servicing station has 2 bays where service can be offered simultaneously. Because of space limitation, only 4 cars are accepted for servicing pattern is Poisson with 12 cars per day. The service time in both the bays is exponentially distributed with mean 8 cars per day per bay. Find the average number of cars in the service station, the average number of cars waiting for service and the average time a car spends in the system.
PROBLEMS UNDER (M/M/1 ):
22. Arrivals at a telephone booth are considered to be Poisson with an average time 12 minutes between one arrival and the next. The length of telephone call is assumed to be distributed exponentially with mean 4 minutes. a) Find the average number of persons waiting in the system. b) What is the probability that a person arriving at the booth has to wait in the queue? c) What is the probability that it will take more than 10 minutes for a person to wait and complete his cell? d) Also estimate the fraction of the day when phone will be in use e) The telephone department will insult a second booth when convinced that an arrival would expect to wait at least 3 minutes for the phone. By how much should flow of arrivals increase in order to justify a second booth? f) What is the average length of queue that forms from time to time?
23. Customers arrive at one man barber shop according to Poisson process with mean interval time of 12 minute. Customer spends an average of 10 minute in the barber chair. a) What is the expected number of customer in the barber shop and in the queue? b) Calculate the percentage of time an arrival can walk straight in to the barber chair without having to wait?
c) How much time can customer expect to spend in the barber’s shop? d) Management provides another chair and hires another barber, when customer waiting time in the shop exceeds 1.25 hrs. How much must the average rate of arrivals increase to warrant a second barber? e) What is the average time customer spends in the queue? f) What is the probability of waiting time in the system is greater than 30 min? g) What is the percentage of customer who has to wait prior getting into the barber chair? h) What is the probability of more than 3 customers in the system?
24. If people arrive to purchase cinema tickets at the average rate of 6 per minute, it takes an average 7.5 seconds to purchase a ticket. If a person arrives 2 min before the picture starts and if it takes exactly 1.5 min to reach the correct seat after purchasing the ticket. a) Can he expect to be seated for the start of the picture? b) What is the probability that he will be seated for the start of the picture? c) How much must he arrive in order to be 99% sure of being seated for the start of the picture?
25. An duplicating machine maintained for office use it’s operated by an office assistant who earns Rs 5 per hour. The time to complete each job varies according to an exponential distribution with mean 6 minute. Assume Poisson input with an average arrival rate of 5 jobs per hour. If an 8-hour day is used as a base Determine a) The percentage idle time of the machine? b) The average time a job is in the system c) The average earning per day of the assistant?
26. In the railway marshalling yard goods trains arrive at a rate of 30 trains per day Assume that inter arrival time follows exponential distribution and the service time distribution is also exponential with an average of 36 minutes. Calculate the following
i) Mean queue size ii) The probability that the queue size is at least 10 iii) If the input of trains increases to an average of 33 trains per day, what will be the change in the above quantities?
27. Customer arrives at one window drive in bank according to a Poisson process with mean 10 per hour. Service time is an exponential distribution with mean 5 minutes. The space in front of the window including that for a serviced car can accommodate a maximum three cars others can wait outside of the space. i) What is the probability that an arriving customer can drive directly to the space in front of the window? ii) What is the probability of arriving customer will have to wait outside of indicated space? iii) How long is an arriving customer expected to wait before being served?
28. At what average rate must be a clerk in a supermarket work in order to ensure a probability of 0.90 that the customer will not wait longer than 12 min ? it is assumed that there is only one customer at which customer arrive in a Poisson fashion at an average rate 15 per hour and length of service is an exponential distribution.?
29. 8. On average of 96 patients per 24 hour day require the service of an emergency clinic. Also an average a patient require 10 minutes of active attention assume the facility can handle only one emergency at a time. Suppose that it cost the clinic Rs 100 per patient treated to obtain an average service time of 10 minutes that each minute of decrease in this average time would cost Rs 10 per patient treated. How much would have to budget by the clinic to decrease the average size of the queue size from patient to ½ patient? 30. 9. A repair man is to be hired to repair machines which breakdown at an average rate of 3 per hour. The break down follows Poisson distribution. Non- productive time of machine is considered to cost Rs 16 per hour. Two repair men have to be interviewed. One is slow but cheap while other is fast but expensive. The slow repairmen charges Rs 8 per hour and he service machine at the rate of 4 per hour . The fast repair man demands Rs 10 per hour and service at the average rate of 6 per hour. Which repairmen should be hired?
PROBLEMS UNDER (M/M/c):
31. A bank has 2 tellers working on savings accounts. The first teller handles with drawls only. The second teller handles deposits only. It has been found that the service time distribution for both deposits and withdrawals are exponentially distributed with mean service time of 3 minutes per customer. Deposits are found to be arriving at a Poisson fashion throughout the day with mean arrival rate of 16 per hour .withdrawals also arrive in a Poisson fashion with mean arrival rate of 14 per hour. What would be the effect on the average waiting time for the customer if each teller handles both withdrawals and deposits? What would be the e ffect if this could only be accomplished by increasing the service time to 3.5 minutes?
32. A supermarket has two girls attending to sales at the counter s. If the service time for each Customer is exponential with mean 4 min and if the people arrive in Poisson fashion at the rate of 10 per hour . a) What is the probability that a customer has to wait for service? b) What is the expected percentage of idle time for each girl? c) if the customer has to wait in the queue, what is the expected length of his waiting time?
33. A petrol pump station has 4 pumps. The service time follows the exponential distribution with mean of 6 min and cars arrive for service in a Poisson distribution at the rate of 30 cars per hour. a) What is the probability that an arrival would have to wait in line? b) Find the average waiting time, average time spent in the system and the average number of cars in the system? c) For what percentage of time would a pump be idle on an average?
34. A telephone company is planning to install telephone booths in new airport. It is established the policy that a person should not wait more than 10% of times he tries to use the phone. The demand for use is estimated to be Poisson with an average of 30 per hour. The average phone call has an exponential distribution with mean time of 5 min. how many phone booths should be installed?
35. 5. There are 3 typists in an office. Each typist can type an average of 6 letters per hour. If letters arrive for being typed at the rate of 15 letters per hour. a) What fraction of time all the typist will busy? b) What is the average number of letters waiting to be typed? c) What is the average time a letter has to spend waiting and for waiting to be typed?
36. An insurance company has 3 claim registers in its branch office. People with claims against the company are found to arrive in an Poisson fashion at an average rate of 20 per 8 hour day . The amount of time that an adjuster spends with a claims is found to have an exponential distribution with mean 40 min .Claimants are processed in the order of their appearance. a) How many hours a week can an adjuster expect to spend with claimants? b) How much time, on average does a claimant spend in the branch office?
37. A telephone exchange has 2 long distance operators. The telephone company finds during the peak load long distance calls arrive in a Poisson fashion at an average rate of 15 per hour. The length of service on these calls in an exponentially distributed with mean length 5 min. a) What is the probability that a subscriber will have to wait for his long distance call during the peak hours of the day? b) If the subscriber will have to wait and are serviced in turn, what is the expected waiting time?
PROBLEMS UNDER (M/M/1):
38. The local one man barber shop accommodates a maximum of 5 people at a time (4 wait and 1 getting hair cut). Customers arrive according to a Poisson distribution with mean 5 per hour. the barber cuts hair at an average rate of 4 per hour .Find a) What percentage of time is the barber idle? b) What fraction of the potential customer turned away? c) What is the expected number of customer waiting for hair-cut? d) How much time can customer expect to spend in the shop?
39. At a railway station, only one train is handled at a time. The railway yard is sufficient only for 2 trains to wait, while other is given signal to leave the station. Trains arrive at an average rate of 6 per hour and the railway station can handled them on an average of 6 per hour. Assume Poisson arrivals and exponential service distribution, Find a) Find the probabilities for number of trains in the system? b) Find the average waiting time of new train coming into the yard? c) If the handling rate is doubled, how will the above results get modified?
40. Patients arrive at a clinic according to a Poisson process at a rate of 30 patients per hour. The waiting room does not accommodated more than 14 patients. Examination time per patient is exponential with mean rate of 20 per hour. a) Find the effective arrival rate at the clinic? b) What is the probability that an arriving patient will not wait? c) What is expected waiting time until a patient is discharged from the clinic? 41. A car park contains 5 cars. The arrival of cars is Poisson at a mean rate 10 per hour. The length of time each car is spends in the car park has negative exponential distribution with mean of 2 min. a) How many cars are in the car park on an average? b) What is the probability of newly arriving customer finding the car park full and leaving to park car elsewhere.
42. A stenographer is attached to 5 officers for who she performs stenographer work. She gets calls from the officers at the rate of 4 per hour and takes on the average 10 min to attend to each call. If arrival rate is Poisson and service time is exponential , find a) the average waiting time for an arriving call b) the average number of waiting calls c) the average time arriving call spends in the system?
PROBLEMS UNDER (M/M/c): 43. A 2 person barber shop has 5 chairs to accommodated waiting customer s. Potential customers who arrive when all the 5 chairs are full ,leave without entering barber shop. Customers arrive at the average rate of 4 per hour and spend an average of 12 min in the barber’s shop. Compute . 44. 2. A car servicing station has 2 bays where service can be offered simultaneously .Because of space limitation; only 4 cars are accepted for servicing. The arrival pattern is Poisson with 12 cars per day. The service time in an exponential distributed 8 cars per day. Find a) The average number of cars in the service station b) The average number of cars waiting for service c) The average time car spends in the system.
45. At a port there are 6 loading berths and 4 unloading crews. When all the berths are full, arriving ships are delivered to an overflow facility 20 kms down to the river. Tankers arrive according to Poisson process with mean 1 every 2 hrs. It takes for an unloading team; the average 10 hrs to unload tanker, the unloading time follows an exponential distribution. Find a) How many tankers are there at the port on the average? b) How long does a tanker spend at the port on the average?
c) What is the average arrival rate at the overflow facility?
46. 4. A group of engineers has 2 terminals available to aid in their calculation. The average computing job require 20 min of terminal time and each engineers require some computation about once every half an hour. Assume that these are distributed according to an exponential distribution. If there are 6 engineers in the group, find a) The exponential number of engineers waiting to use one of the terminals and in the computing centre b) The total time lost per day
