5. Queue (aiting Line) Queuing terminology Input Service Output To decribe a ueuing ytem, an input proce and an output proce ha to be pecified. For example ituation input proce output proce Bank Cutomer arrive Teller erve at bank the cutomer Input (arrival proce): Arrival are called cutomer. e aume that at mot one arrival occur at a given time (if more than one arrival can occur we ay that bulk arrival are allowed). Uually the arrival proce i unaffected by the number of cutomer preent in the ytem. There are, however, ituation in which the arrival proce depend on the number of cutomer preent. For example, -finite ource model - arrival are drawn from the mall population - balking - a cutomer arrive but fail to enter the ytem (facility i too crowded). Service mechanim (output): To decribe the output proce we pecify: ) Number of erver (channel) ) Arrangement of erver: erver in parallel (for example bank) Parallel erver in erie (for example an aembly line) Serie
3) Number of line line line Jockeying - changing line 4) Service time Queue Dicipline The ueue dicipline decribe ) the order in which cutomer are erved FCFS (firt come, firt erved) LCFS (lat come, firt erved) SIRO (ervice in random order) ) limit on ueue length Reneging - leave ueue before ervice. The Kendall -Lee Notation for Queueing Sytem Each ueuing ytem i decribed by ix characteritic: Input proce/service ditribution/number of erver/ueue dicipline/max number of cutomer in the ytem/ize of population In many important model 4/5/6 i GD/ /. If thi i the cae, then 4/5/6 i often omitted. M - i exponentially ditributed D - determinitic G - general ditribution of ervice time GI - general independent ditribution of inter-arrival time GD general ueue dicipline Sytem i ueue plu ervice facility. The baic M/M/ model M - inter-arrival time and ervice time are exponentially ditributed There i one erver in the ytem e aume that at mot one arrival can occur at a given intant of time. Let t i be the time at which the ith cutomer arrive. Ti ti+ t i i the ith inter-arrival time.
e aume that all the T i ' are independent, continuou random variable decribed by the random variable A with the probability denity function at () e t. The mean or average inter-arrival time i eual to /: t ta( t) dt te dt Similarly variance of inter-arrival time Hence i the arrival rate ( unit of arrival per unit time) Probability of next arrival in [,T] T t T e dt e o Probabilit y of no arrival in[,t] e T Service time probability denity function i µ µ t, o mean ervice time i /µ... : arrival rate, µ: ervice rate e /µρ the utiliation factor Aume ρ< (teady tate ditribution exit) Suppoe 6, µ4 cutomer per hour- after a long time number of cutomer in the ytem would "blow up". Quantitie of interet: )The average number of cutomer preent in -the ueuing ytem -ervice -ueue L L L Check : L L + L µ ρ ρ ρ
) The amount of time that a typical cutomer pend in - the ytem -line -ervice L L L 3) The teady-tate probability that j cutomer will be preent Example j ρ ( ρ) j An average of car per hour arrive at a ingle erver drive-in teller. The average ervice time for each cutomer i 4 minute, and both inter-arrival time and ervice time are exponential. e are dealing with M/M/ ueuing ytem,, µ5. Hence, ρ/3. hat i the probability that the teller i idle? ( o ρ ) 3 hat i the average number of car waiting in line for the teller? e eek L ρ ρ 4 3 hat i the average time a cutomer pend in the ytem? e eek L ρ ( ρ) 5 Thu, minute.
A ueueing optimization model Queueing theory can be ued a an aid in deciion making. Problem in which a deciion maker mut chooe between alternative ueuing ytem are called ueuing optimization problem. Let ay that we want to minimize the um of the of the hourly ervice cot and the expected hourly cot due to the delay cot. Expected cot/hour ervice cot/hour + expected delay cot/hour The computation of the hourly ervice cot i uually imple. To find the hourly delay cot we notice that expected delay cot/hour (expected delay cot/cutomer) (expected number of cutomer/hour) M/M/ Interarrival time are exponential, with rate, Service time are exponential, with rate µ, There i a ingle line of cutomer waiting to be erved at one of parallel erver. e define ρ. µ e have analogou but more complex formulae for j, L, L, L,,, : (ρ< ) ρ ( ) i! i i ( ρ) j i ( ρ) +!( ρ) j j,, K, j! The teady tate probability that all erver are buy i given by P ( j ) ( ρ)! ( ρ) Table 6/p4 tabulate P(j ) for variety of ituation.
L µ! + µ P ( j ) ρ ρ L L L + L µ µ L M/G/ Single erver ueuing ytem, Exponential interarrival time with arrival rate Arbitrary ervice time with mean /µ, and variance σ. It can be hown that : L σ + ρ ( ρ ) where ρ µ L ρ L L µ L + µ + Queue in erie e conider the ituation when the cutomer i erved by more then one erver. ρ Input Output... The cutomer after completing the firt ervice, wait for and undergoe the econd ervice... If interarrival time for a erie ueuing ytem are exponential with rate, ervice time for each tage are exponential, and each tage ha an infinite capacity waiting room, the interarrival time for arrival to each tage of the ueuing ytem are exponential with rate. (Jackon 957)
Example (inton page 5 [55]) The lat two thing that are done to a car before it manufacture i complete are intalling the engine and putting on the tire. An average of 54 car per hour arrive reuiring thee two tak. One worker i available to intall the engine and can ervice an average of 6 car per hour. After the engine i intalled, the car goe to the tire tation and wait for it tire to be attached. Three worker erve at the tire tation. Each work on one car at a time and can put tire on a car in an average of 3 minute. Both interarrival and ervice time are exponential. ) Determine the mean ueue length at each work tation. ) Determine the total expected time that a car pend waiting for ervice. Solution: Stage Stage Tire Engine Tire Tire Thi i a erie ueuing ytem with 54 car per hour,, µ 6 car per hour, 3, µ car per hour. Since < µ, and < 3µ, neither ueue will "blow up", and Jackon' theorem i applicable. For tage : 54 ρ.9 6 ρ L ( for engine ) 8. car ρ And For tage : L ( for engine ). 5 hour ρ 54.9 3
3 9 3 9 (. ) (. ) + i! 3!(. 9) i i 3 i 3+ 54 L ( tire ) 7. 47 car 54 3 3! 3 L ( for tire). 38hour Thu, the total expected time a car pend waiting for engine intallation and tire i.5+.38.88 hour. Tutorial Problem.4., 3, 4, 8, 9,, pp 8- [.4., 3, 4, 5, 6, 8, 9 pp 33-34].review., pp 843-44 [.review., 8 pp 8-8]