SINGLE-CHANNEL QUEUING ROBLEMS AROACH Abdurrzzag TAMTAM, Doctoral Degree rogramme () Dept. of Telecommuicatios, FEEC, BUT E-mail: xtamta@stud.feec.vutbr.cz Supervised by: Dr. Karol Molár ABSTRACT The paper deals with oe problem of sigle chael queuig problems. It shows that sigle-chael queuig model without losses ca be used to solve queuig problems. It is composed of the queue ad service ceter ad uses the FIFO system. The statio performs the service for the first elemet which appears i lie. INTRODUCTION I order to describe queuig problems through mathematical formulatio, some assumptios are made by cosiderig arrivals ad service as pattered by ow fuctio. Equatios represetig the distributio of the time betwee arrivals are used with other equatios depictig other features such as the distributio of the service time. The relatioship existig betwee these equatios is the matter studied i waitig lie theory. Arrivals of people or etry requiremets (evets) are customarily oisso distributed. The duratio of the service provided by people is usually expoetially distributed. SINGLE-CHANNEL QUEUING ROBLEMS Sigle-statio or sigle-chael queuig problem is the ame applied o those problems i which oly oe uit (statio) is deliverig the service as illustrated i Fig., where circles represet the arrival elemets (evets) ad a square represets a statio which cotais a elemet beig serviced. Fig. : Sigle-chael queuig problem. OISSON ARRIVALS The oisso is a discrete probability distributio ad yields the umber of arrivals i a give time. The expoetial distributio is a cotiuous fuctio ad yields the distributio of
the time itervals betwee arrivals. The oisso distributio cosider the behavior of arrivals as occurrig at radom ad postulates the presece of a costat which is idepedet of the time. The costat represets the mea arrival rate or the umber of arrivals per uit of time, ad is the legth of the time iterval betwee two cosecutive arrivals. The oisso distributio is expressed by the followig formula: ( T ) ( T )! e T T,,... < where the parameter is the probability of the arrival which occurs betwee the time t ad t+ t, ad e is the base of the atural system of logarithms. The expected umber of arrivals through the iterval (, T) is T. We calculate the mea value µ of arrivals µ ( T ) ( t) T T ( T ) e e T!! e T t e T T By assigig the value to the period i.e. T, we have e,,...! T The probability that per the iterval (,T) does ot come ay evet is ( T ) e T The complemetary situatio is described as follows: ( T ) ( T ) e The differece i the last formula is the distributio fuctio of the expoetial distributio with the fuctio desity equal to e T, so the mathematical expressio of the distributio fuctio is the F ( T ) ( τ T ) e T T < T The fuctio F(T) has this sese: It is the probability that the time iterval betwee two cosecutive arrivals will be equal or less tha the value of T. We calculate the mea time T T betwee two arrivals. E ( τ ) T d F( T ) T e dt [ e ].. EXONENTIAL SERVICE TIMES Whe the servicig of a uit taes place betwee time t ad t (for t sufficietly small) the service times are give by expoetial distributio, while the service rates are give by the oisso distributio. The parameter µ T idicates that µ is a costat of proportioality, which is idepedet of time, of the queue legth, or of the features. Agai, callig the umber of potetial services which ca be performed i the iterval (, T ),the oisso formula for the servicig rate is ( ) ( µ T ),,... µ T p T e! T < The mea servicig rate which is the expected umber of services performed i oe uit of time) is idicated by µ whe the servicig time is expoetial. It ca be foud approximately by dividig the output (time) of the services delivered alog the period T. by the portio of T i which the services are really operatig. The mea servicig time MST is the reciprocal of µ, MST µ
.3 SINGLE-CHANNEL QUEUING MODEL WITHOUT LOSSES (M/M/) Customarily, the iputs, as well as the legth of time required by the statio to perform the requested wor, are cosidered to arrive at radom. The servicig rate is idepedet of the umber of elemets i lie. The statio performs the service for the first elemet which appears i lie (FIFO System First I, First Out). Whe the service is busy, the icomig elemet waits i lie i order of arrival util the previous elemet leaves the chael at the ed of its service. We suppose a ifiite source of arrival elemets. This system is ofte called the system of bul service (SBS). It is composed of the queue ad service ceter. We put a list of otatios which will be useful i the ext mathematical cosideratios: mea arrival rate (umber of arrivals per uit of time) µ mea service rate (per chael) umber of elemets i SBS t probability that a ew elemet eters the SBS betwee t ad t + t time iterval µ t probability that a elemet has received service (completely fiished) betwee t ad t + t time iterval - t probability of havig o arrivals i the iterval (t, t + t ) -µ t probability of havig o elemets serviced durig the iterval (t, t + t ) + (t) probability of havig + elemets i the SBS system at time t - (t) probability of havig - elemets i the SBS system at time t (t + t ) probability of havig elemets i the SBS system at time t Let us suppose >. We calculate the probability ( t + t ) (t + t ) (t) (- t) (-µ t) + + (t) (µ t) (- t) + - (t) ( t) (-µ t) + + (t) ( t) (µ t) for >.. ( ) t is a very small iterval hece we ca omit its square i.e. we approximate it by zero. We receive uder give supposed coditio from ( ) the followig system of equatios: (t + t ) (t) + (µ t) + (t) + ( t) + (t) - (( + µ ) t ) (t) > (t + t ) - (t) (µ t) + (t) + ( t) + (t) - (( + µ ) t ) (t) > we divide the equatio by t ad hece ( t + t) ( t) t µ + (t) + + (t) - ( + µ ) (t) ( ) whe t approaches zero, the followig differetial equatios ca be stated: ( t) d d t µ + (t) + + (t) - ( + µ ) (t) ( 3 ) which expresses the relatioship amog the probabilities, -, + at the time t ad the mea arrival rate ( ) ad the mea service rate ( µ ). The probability that o elemets will be i the (SBS) is give by the equatio ( 3 ).
d ( t ) - ( t ) + µ ( t ) ( 4 ) d t We suppose that the system is worig for a ubouded time iterval ad that it pass to the steady state-coditio. I this momet ( t ), which are probabilities which were be depedet o time t they become idepedet accordig to time ad thus d for,,, 3,... ( 5 ) d t Thus ( 3 ) ad ( 4 ) are trasformed o homogeous liear equatios : µ + + - - ( + µ ),, 3,... ( 6 ) - + µ, ( 7 ) where,, 3,...,,... are uow values. We have from ( 7 ) ad after a rewritig of the equatio ( 6 ) for we have µ µ + - ( + µ ) Hece µ ( + µ ) - ad after the substitutio for we have : µ hece usig the iductio we receive the commo formula for the probability that the sigle-chael system cotais together just (i the queue i the lie ad i the service ceter ) arrived elemets (demads). µ ( 8 ) We deote the ratio µ by. We call it traffic itesity, which is the expected service per uit of time measured i erlags (i hoor of A.K.Erlag, who is cosidered the father of the queuig theory). For ext we suppose that <. Now we use the fact that ad we substitute. ( + + +... + +... ) hece... + + + which is the geometric progressio with the quotiet < We have for idividual cases: ( ),, 3,..., hece. ( 9 ) ( ) The equatio () expresses the probability of existece of waitig queue of the legth for >. (Note that this equatio is valid, as already idicated, oly whe < µ ).
The average umber of elemets (evets), both waitig i the queue ad atteded i service is: ( ) ( ) m ( ), ( ) after rearragemets we obtai m. µ The ifiite series is of expoetial type. Hece we ca apply the followig rule: ( ) S d d, usig this formula we receive: ( ) S d But we eed the value of S() which is the derivative of the fractio. The mea legth (m Q ) of the queue (of the waitig elemets excludig the elemet uder the service process is obtaied from the defiitio of the mea value i theory of probability ad we have: m Q ( ) m ( ) m. µ µ The mea time betwee arrivals ( h ), where the arrivals are oisso distributed, is obtaied by reciprocatig the mea arrival rate, hece h. The average time of demurrage of the elemet i all the system, i.e. i queue ad i service (WTS ) is expressed i terms of ad µ as follows: WTS m. µ µ The average waitig time of a elemet i queue (WTQ) is expressed i terms of ad µ as follows: m WTQ Q µ µ µ µ µ ( ) 3 CONCLUSIONS The former model admits the queues of arbitrary legths. Equatio () gives the probability that a elemet will ot have to wait at all upo its arrival at the service statio before goig ito oe-chael service. A model with queues of arbitrary legth is ofte called a model without losses. REFERENCES [] Acoff, R., Sassie, M.: Fudametals of Operatios Research. N. Y. Wiley, 968. [] Bellma, R., Dreyfus, S.: Dyamic programmig. rticeto, U, 96. [3] Dudori, J.: Operačí výzum. raha, ČVUT, 997. [4] Saaty, T.: Mathematical Methods of Operatios Research. N.Y Mac Grave, 959. [5] Zapletal, J.: Operačí Aalýza. VOŠ Kuovice, 995.