LECTURE - 5 M/M/1/N QUEUING MODEL. Learning objective To discuss single server finite queues (M/M/1/N queues)

Transcription

1 ECTURE - 5 M/M// QUEUIG MODE earig objective To discuss sigle server fiite queues (M/M// queues)

2 9.8 Queuig model with a fiite queue I real cases, queues ever become ifiite, but are limited due to space, time or service operatig policy. uch queuig model falls uder the category of fiite queues. Examples: arkig of vehicles i a supermarket is restricted to the space of the parkig area imited seatig arragemet i a restaurat Fiite queue models restrict the umber of customers allowed i service system. That meas if represets the maximum umber of customers allowed i the service system, the the () th arrival will depart without beig part of the service system or seekig service robability of havig o customers i M M Queuig Model The service system ca accommodate customers oly. The ()th customer will ot joi the queue We kow that, probability of havig oe customer i the queuig system is as metioed below. λ µ imilarly, probability of havig customers i the service system will be λ µ ice it is fiite queuig model, the sum of probabilities till th customer will be as writte below.

3 λ µ λ µ λ where represets µ Hece, - - r obability of havig o customer i the fiite si gle server queue is fiite Geometric r ogressio series robability of havig customers i fiite queuig model Due to restrictio i the allowable umber of customers i the system, the service system will be iterested to kow the expected umber of customers who are lost. This ca be foud out by determiig the probability of customers ot joiig the system. I represets the probability of ot joiig the system, the the arrival rate (λ) times will represet the expected umber of customers who are lost after the service system becomes full. Fiite Queues ca be M M type or M M c type. I M M type queuig model, the utilizatio factor, ca exceed uity. Also, i a sigle server case, - represets the maximum umber of customers i the queue. I M M c type queuig model, must be equal to or greater tha the total umber of servers c.

4 A arrivig customer will ot be etertaied or allowed i the system if the umber of customers i the system equals or the legth of the queue is -c erformace measures of M M queuig system Average umber of customers i the system, s ca be determied usig probability of havig fiite,, customers i the service system. We kow that for a fiite queue system, probability of havig customers with (>) ca be writte as (>) - Hece, the umber of customers i queue ca determied as give below. q s (- ) - ( ) ( ) ( ) ( ) ( ) ( )( ) s

5 As metioed i sectio 9.8.2, represets the probability for the customers of ot joiig the system. Hece, the λ represets the average customers lost due to fiite queue. o, the effective arrival rate, λ e, i such queues is represeted as give below: λ e λ- λ or λ e λ(- ) Usig effective arrival rate ad ittle s law we ca determie other performace measures of M M queuig system, which are give below. Average waitig time i the queuig system, W s, comes out to be as metioed below. W q λ( ) µ Average waitig time i the queue, W q, ca be writte as give below. W q W Example µ The arrival rate of cars at GRAD jewelers is 2 cars per hour. The service rate at jeweler shop is 8 cars per hour. The oly parkig area at GRAD jewelers is restricted to 6 cars oly. The arrival rate ad service rate follows oisso distributio. Idetify the type of queue ad the determie the performace measures of this queue. Aalyze the result if service rate goes to 2 cars per hour. olutio ice, arrival ad service rate follows oisso distributio ad there is oe parkig area with restricted capacity of cars, this is

6 M M Queue Arrival rate, λ 2 cars per hour ervice rate, µ 8 cars per hour 6cars 2 Utilizatio ratio (.) 7 (.) (.9) ( ) ( ) ( ) Give,. 6 7 (7)(.) 6(.). 7 (.) (.) cars λ λ( ) e 2(.97) 6.66

7 ( ) q. (.).9 7 q W W q 3.4 (.9) 2.52 cars q λ µ e hour or 2.66 mi utes W µ hour or 9.32 mi utes If the service rate becomes 2 cars per hour the; µ λ hece, cabe writte as givebelow cars

8 Hit Wekow for fiiteque, which ca be writte as... 2 If the, substitute [ ] ( ) Utility of Queuig models i service capacity plaig ervice orgaizatio faces a commo trade-off betwee cost of providig service ad cost of customer waitig. The queuig models ca help i determiig the desired service level which ca maage these two types of costs. 9.9.(a) Determie the service capacity for desired expected customer waitig time. The service orgaizatio ca utilize MMcmodel; that is multiple servers queuig model to determie the service capacity. If we set a specific time of waitig i.e. customers should wait o more tha the specific time. ( W q kow), we ca determie other performace measures icludig optimal umber of servers (b) Determie the umber of servers based o the probability that X or more percet of all customers should experiece a delay of less that D time uits.

9 We will agai take help of MMcmodel. I MMcqueue model, we kow that probability of havig customers is as give below λ! if c µ λ λ >! if c c µ cµ C c We ca utilize the above relatio to determie ( c ) After writig the expressio for ( c, ) we ca keep varyig the value of c till we achieve the probability of fewer tha X percet of arrivig customers have to wait.