Queuig Theory Prelimiaries A flow of customers comig towards the service facility forms a queue o accout of lack of capacity to serve them all at a time. RK Jaa Some Examples: Persos waitig at doctor s cliic Persos waitig at railway bookig office Machies waitig to be repaired Ships waitig i the harbor to be uloaded Airplaes take off, ladig 1 Customers may be: persos, machies, vehicles, parts etc 2 Applicatios of Queuig Theory Basic Cocepts Telecommuicatios Traffic cotrol Determiig the sequece of computer operatios Predictig computer performace Health services (eg. cotrol of hospital bed assigmets) Airport traffic, airlie ticket sales Layout of maufacturig systems Customers Iput Process Queue Service Mechaism Queuig system: Customers arrivig for service, waitig for service if it is ot immediate, leavig the system after beig served. 3 The theoretical study of waitig lies, expressed i mathematical terms. 4 Iput Process Queue Disciplie Source (populatio size) Queue size Fiite Fiite Ifiite Ifiite Arrival behavior Queue disciplie Oe by oe First come, first served (FCFS) Batch Fixed size Variable size Chage of arrival patter Last come, first served (LCFS) Radom selectio for service (RSS) Priority queue Preemptive o-preemptive Statioary o-statioary/trasiet (time depedet) 5 6 1
Service Mechaism umber of Service Chaels Bulk service (computer with parallel processig, bus service) State depedet (service depeds o umber of waitig customers. Example: iteret access) State idepedet Statioary/o-statioary Sigle chael Parallel chaels (provides idetical service) Series (customers go through a umber of services, public offices, maufacturig process) 7 8 Capacity of the System QT i Performace Measuremet Fiite source queue (fiite capacity of waitig room, restrictio o queue legth) Ifiite (o restrictio o queue legth) Aalyze waitig time distributio To kow average waitig time of customer To kow queue legth distributio Calculate curret work backlog Measuremet of the idle time of server Measuremet of the busy time of server System utilizatio 1 Questios I Queuig Systems otatios 1. The umber of people i the system (those beig served ad waitig i lie). 2. The umber of people i the queue (waitig for service). 3. The waitig time i the system (the iterval betwee whe a idividual eters the system ad whe he or she leaves the system). 4. The waitig time i the queue (the time betwee eterig the system ad the begiig of service). : # customers i the system (i queue & i service) P (t): Trasiet state probability of exactly customers i the system at time t (it is assumed that the system starts its operatio at time zero) P : Steady state probability of exactly customers i the system E(s): Expected umber of customers i the system E(q): Expected umber of customers i the queue E(s) umber of customers beig served 11 12 2
otatios: Cotiued The Poisso Process λ : Expected umber of arrivals per uit time (mea arrival rate) of customers whe customers are preset i the system µ : Expected umber of customers served per uit time (mea service rate) whe customers are preset i the system λ: Mea arrival rate whe λ is costat for all µ: Mea service rate whe µ is costat for all 1 E(w1): Expected waitig time per customer i the system E(w2): Expected waitig time per customer i the queue 13 Axiom 1: The umber of arrivals i o-overlappig itervals are statistically idepedet Axiom 2: The probability of more tha oe arrival betwee time t ad time (t + t) is O( t) i.e., the probability of more two or more arrivals durig the small time is egligible. Axiom 3: The probability that a arrival occurs betwee time t ad time (t + t) is {λ. t + O( t)} 14 The Arrival Theorem The Role of Expoetial Distributio If the arrivals are completely radom, the the probability distributio of the umber of arrivals i a fixed time iterval follows Poisso distributio. λt e ( λt) P ( t),,1, 2,! Most aalytic results for queuig situatios ivolve the expoetial distributio either as the distributio of iter-arrival times or service times or both. The followig three properties help to idetify the set of circumstaces i which it is reasoable to assume that a expoetial distributio will occur. - Lack of memory - Small service times - Relatio to the Poisso distributio 15 16 Lack of Memory Small Service Time I a arrival process, this property implies that the probability that a arrival will occur i the ext few miutes is ot iflueced by whe the last arrival occurred. (a) There are may idividuals who could potetially arrive at the system (b) Each perso decides to arrive idepedetly of the other idividuals (c) Each idividual selects his or her time of arrival completely at radom 17 Prob S < t 1..632 t 1 2 3 4 This graph shows the probability that the service time S is less tha or equal to t if the mea service time is 1. 18 3
Cotiued Relatio with Poisso Distributio The graph showed that more tha 63% of the service times were smaller tha the average service time (1). Compare this to the ormal distributio where oly 5% of the service times are smaller tha the average. The practical implicatio is that a expoetial distributio ca best be used to model the distributio of service times i a system i which a large proportio of jobs take a very short time ad oly a few jobs ru for a log time. If the time betwee arrivals has a expoetial distributio with parameter, the i a specified period of time the umber of arrivals will have a Poisso distributio. 1 2 Distributio of Iter-arrival Time The Mea Arrival Time Let T be the time betwee two cosecutive arrivals. If the arrivals o -customers i time t follows Poisso distributio the T follows expoetial distributio. t λe λ, t < f ( t), elsewhere E( T ) tf ( t) dt t t 1 λ λte λt dt 21 22 Distributio of Service Time The Traffic Itesity If T be the iter-arrival time, the the probability of - complete service i time T is give by: φ ( t) P( -servicei time T ) µ t e ( µ t),,1,2,! For Poisso arrival ad departure with oe server, the traffic itesity () is give by: mea arrival rate λ mea service rate µ 23 The uit of is Erlag 24 4
Queuig Models (M / M / 1) : ( / FCFS / ) The geeral model ca be completely represeted by Kedall s otatio as follows: (a / b / c) : (d / e / f) a arrival distributio d capacity of the system b service distributio e service disciplie c # service chaels f size of callig source Stadard otatios: M Poisso arrival or departure distributio E k Earlagia or Gamma arrival or departure distributio GI Geeral Idepedet arrival distributio 25 Sigle chael ifiite populatio model I a steady state coditio < 1, it ca be show that λ P 1 1, < 1 µ λ λ P 1 (1 ), < 1, µ µ 26 Characteristics of the Model Cotiued 1. Expected umber of customers i the system E(): E( ) P (1 ) (1 ) d d (1 ) ( ) (1 ) d d 1 d 1 (1 ) λ (1 ) 2 d (1 ) (1 ) 1 µ λ 27 2. Expected queue legth E(q): Sice there is oe server, oe customer is i service & (-1) customers are i the queue. E( q) ( 1) P P P 1 1 1 P P P {1 (1 )} 1 2 2 λ, sice P (1 ) 1 1 µ ( µ λ) 28 Cotiued Cotiued 3. Probability of queue size greater tha some fiite umber : P(Queue size ) P (1 ) r (1 ) (1 ), where r r 1 λ (1 ) (1 ) µ 2 4. Expected waitig time per customer i the system E(w 1 ): Expected # customers i the system E( ) 1 1 E( w1 ) Arrival rate λ µ λ µ (1 ) 5. Expected waitig time per customer i the queue E(w 2 ): 1 1 E( w2 ) E( w1 ) service time of oe customer µ λ µ µ (1 ) 3 5
Example: 1 Solutio: 1 I a railway marshallig yard, goods trais arrive at a rate of 3 trais per day. Assumig that the iterarrival time follows a expoetial distributio ad the service time (time take to hump a trai) distributio as Poisso with a average of 36 miutes, calculate: (i) The average umber of trais i the system. (ii) The probability that the queue size exceeds 1. (iii) Expected waitig time i the queue. (iv) Average umber of trais i the queue. 31 λ 3/(24 6) 1/48 trais/mi µ 1/36 trais/mi λ/ µ 36/48.75 (i) Average umber of trais i the system / (1 - ).75/(1.75) 3 trais 1 1 (ii) CP (Queue size 1) (.75).6 (iii) Expected waitig time i the queue: 18 mi 1 µ (1 ) hr 48 mi (iv) Average umber of trais i the queue: 2 2.25 i.e., early 2 trais 1 32 Example: 2 Solutio: 2 I a sigle serve system, the arrival rate λ 5 per hour ad the service rate µ 8 per hour. Assumig the coditios of for the sigle chael queuig model, fid out: (i) The probability that the system is idle. (ii) The probability that the queue size is at least 2. (iii) Expected time that a customer is i the queue. (iv) The probability that a customer is beig served ad obody is waitig. (v) Expected time a customer speds i the system. (vi) Average umber of customers i the queue. 33 λ 5 per hr, µ 8 per hr λ/µ 5/8.625 (i) Probability that the system is idle P (o customer i the system) 1 λ/µ (ii) P (At least 2 customers i the system) P( 2) (λ/µ) 2 (iii) Expected time a customer is i the queue: λ/µ(µ -λ) (iv) The probability that a customer is beig served ad obody is waitig P 1 (1 λ/µ) λ/µ (1 ) (v) Expected time a customer speds i the system 1/(µ - λ) 2 2 (vi) Average umber of customers i the queue λ µ ( µ λ) 1 34 (M / M / 1) : ( / FCFS / ) Cotiued Sigle chael fiite populatio model Maximum umber of customers allowed i the system: Maximum queue legth: ( 1) customers are i the system: o ew arrival is permissible I a steady state coditio < 1, it ca be show that I this model, λ/µ may be or 1 for steady state coditio. Because, the umber of customers allowed i the system is cotrolled by the queue legth ad ot by the relative rates of arrival (λ) or departure (µ). 1, 1 1 1 + P 1, 1 + 1 P (1 ), 1 + 1 1 1, 1 + 1 35 36 6
Characteristics of the Model Characteristics of the Model 1. Expected umber of customers i the system E(): + 1 1 ( + 1) +, 1 + 1 (1 )(1 ) E( ), 1 2 2. Expected queue legth E(q): + 1 1 ( + 1) + (1 ), 1 + 1 + 1 (1 )(1 ) + (1 ) E( q) 1 + 1, 1 2 + 1 37 38 Cotiued Example: 3 3. Expected waitig time per customer i the system E(w 1 ): 1 1 +, 1 µ (1 ) (1 ) E( w1 ) + 1, 1 2λ 4. Expected waitig time per customer i the queue E(w 2 ): 1 1 1, 1 µ + (1 ) (1 ) E( w2 ) + 1 1, 1 2λ µ 3 Assume that the goods trais are comig i a yard at the rate of 3 trais per day ad suppose that the iter-arrival time follows a expoetial distributio. The service time for each trai is assumed to be expoetial with a average of 36 miutes. If the yard ca admit trais at a time, the calculate: (i) The probability that the yard is empty. (ii) The expected umber of trais i the yard. (iii) The expected umber of trais i the queue. (iv) Expected waitig time of a trai i the yard. (v) Expected waitig time of a trai i the queue. 4 Solutio: 3 Solutio 3: Cotiued λ 3/(24 6) 1/48 trais/mi µ 1/36 trais/mi λ/ µ 36/48.75 1 (i) Probability that the yard is empty 1 P for 1 1 1.75.28 1 (.75) 41 (ii) The expected umber of trais i the yard: + 1 1 ( + 1) + E( ), 1 + 1 (1 )(1 ) 1.75 1 1(.75) + (.75) (1.75){1 (.75) 1 } 42 7
Solutio 3: Cotiued Solutio 3: Cotiued (iii) Expected umber of trais i the queue: + 1 1 ( + 1) + (1 ) E( q), 1 + 1 + 1 (1 )(1 ) + (1 ) + 1.75 1 1(.75) + (.75) (1.75) (1.75){1 (.75) 1 } {1 (.75) 1 } 43 (iv) Expected waitig time of a trai i the yard: 1 1 E( w1 ), 1 µ + (1 ) (1 ) 1 (.75) 36 + (1.75) {1 (.75) } (v) Expected waitig time of a trai i the queue: 1 1 E( w2 ) 1, 1 µ + (1 ) (1 ) 1 (.75) 36 + 1 (1.75) {1 (.75) } 44 8