A. Description: A simple queueing system is shown in Fig Customers arrive randomly at an average rate of

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1 Queueig Theory INTRODUCTION Queueig theory dels with the study of queues (witig lies). Queues boud i rcticl situtios. The erliest use of queueig theory ws i the desig of telehoe system. Alictios of queueig theory re foud i fields s seemigly diverse s trffic cotrol, hositl mgemet, d time-shred comuter system desig. I this chter, we reset elemetry queueig theory. QUEUEING SYSTEMS A. Descritio: A simle queueig system is show i Fig. 6-. Customers rrive rdomly t verge rte of. Uo rrivl, they re served without dely if there re vilble servers; otherwise, they re mde to wit i the queue util it is their tur to be served. Oce served, they re ssumed to leve the system. e will be iterested i determiig such qutities s the verge umber of customers i the system, the verge time customer seds i the system, the verge time set witig i the queue, etc. Arrivls Queue Service Dertures Fig.6- A simle queuig system The descritio of y queueig system requires the secifictio of three rts:. The rrivl rocess. The service mechism, such s the umber of servers d service-time distributio 3. The queue discilie (for exmle, first-come, first-served) B. Clssifictio : The ottio A/B/s/K is used to clssify queueig system, where A secifies the tye of rrivl rocess, B deotes the service-time distributio, s secifies the umber of servers, d K deotes the ccity of the system, tht is, the mximum umber of customers tht c be ccommodted. If K is ot secified, it is ssumed tht the ccity of the system is ulimited. Exmles: M/M/ queueig system (M stds for Mrkov) is oe with Poisso rrivls, exoetil service-time distributio, d servers. A M/G/l queueig system hs Poisso rrivls, geerl service-time distributio, d sigle server. A secil cse is the M/D/ queueig system, where D stds for costt (determiistic:) service time. Exmles of queueig systems with limited ccity re telehoe systems with limited truks, hositl emergecy rooms with limited beds, d irlie termils with limited sce i which to rk ircrft for lodig d ulodig. I ech cse, customers who rrive whe the system is sturted re deied etrce d re lost.

2 C. Useful Formuls Some bsic qutities of queueig systems re L: the verge umber of customers i the system L q : the verge umber of customers witig i the queue L s : the verge umber of customers i service : the verge mout of time tht customer seds i the system q : the verge mout of time tht customer seds witig i the queue s : the verge mout of time tht customer seds i service My useful reltioshis betwee the bove d other qutities of iterest c be obtied by usig the followig bsic cost idetity: Assume tht eterig customers re required to y etrce fee (ccordig to some rule) to the system. The we hve: Averge rte t which the system ers x verge mout eterig customer ys (6.) where, is the verge rrivl rte of eterig customers X ( t) lim t t d X(t) deotes the umber of customer rrivls by time t. If we ssume tht ech customer ys $ er uit time while i the system, the Eq. 6. yeilds: L x (6.) Equtio (6.) is sometimes kow s Little's formul. Similrly, if we ssume tht ech customer ys $ er uit time while i the queue,the Eq. 6. yields x w q (6.3) Lq If we ssume tht ech customer ys $ er uit time while i service, Eq. (6.) yields Ls x w s (6.4) Note tht Eqs. (6.) to (6.4) re vlid for lmost ll queueig systems, regrdless of the rrivl rocess, the umber of servers, or queueig discilie. BIRTH-DEATH PROCESS e sy tht the queueig system is i stte S, if there re customers i the system, icludig those beig served. Let N(t) be the Mrkov rocess tht tkes o the vlue whe the queueig system is i stte S, with the followig ssumtios:. If the system is i stte S, it c mke trsitios oly to S -, or S +,, ; tht is, either customer comletes service d leves the system or, while the reset customer is still beig serviced, other customer rrives t the system ; from S o, the ext stte c oly be S.. If the system is i stte S, t time t, the robbility of trsitio to S +, i the time itervl t. e refer to s the rrivl rmeter (or the birth rmeter). (t, t + Δ t) is Δ 3. If the system is i stte S, t time t, the robbility of trsitio to S -, i the time itervl Δ Δ (t, t + t) is d t. e refer to d s the derture rmeter (or the deth rmeter). The rocess N(t) is sometimes referred to s the birth-deth rocess.

3 Let (t) be the robbility tht the queueig system is i stte S, t time t; tht is, (t) P{N(t) } The stte trsitio digrm for the birth-deth rocess is show i Fig. 6-: d d d 3 d d + here 0 0 d 0 0 dd 0... d d... d 0 THE M/M/ QUEUEING SYSTEM I the M/M/ queueig system, the rrivl rocess is the Poisso rocess with rte (the me rrivl rte) d the service time is exoetilly distributed with rmeter (the me service rte). The the rocess N(t) describig the stte of the M/M/ queueig system t time t is birth-deth rocess with the Followig stte ideedet rmeters: The where 0, 0, d, ( ) <, which imlies tht the server, o the verge, must rocess the customers fster th their verge rrivl rte; otherwise the queue legth (the umber of customers witig i the queue) teds to ifiity. The rtio The verge umber of customers i the system is give by is sometimes referred to s the trffic itesity of the system. L 3

4 ( ) q ( ) ( ) L q ( ) Exmles:. Customers rrive t wtch reir sho ccordig to Poisso rocess t rte of oe er every 0 miutes, d the service time is exoetil r.v. with me 8 miutes. () Fid the verge umber of customers L, the verge time customer seds i the sho, d the verge time customer seds i witig for service q. (b) Suose tht the rrivl rte of the customers icreses 0 ercet. Fid the corresodig chges i L,, d q. () The wtch reir sho service c be modeled s M/M/ queueig system with /0 & /8. Thus, we hve L / 0 / 8 / 0 / 8 /0 q s miutes (b) Now /9 & /8 L / 9 / 8 / 9 / 8 / 9 q s miutes Miutes It c be see tht icrese of 0 ercet i the customer rrivl rte doubles the verge umber of customers i the system. The verge time customer seds i queue is lso doubled.. A drive-i bkig service is modeled s M/M/ queueig system with customer rrivl rte of er miute. It is desired to hve fewer th customers lie u 99 ercet of the time. How fst should the service rte be? P( or more customers i the system} ( ) I order to hve fewer th customers lie u 99 ercet of the time, we require tht this robbility be less th 0.0. Thus,

5 from which we obti or. 04 Thus, to meet the requiremets, the verge service rte must be t lest.04 customers er miute. 3. Peole rrive t telehoe booth ccordig to Poisso rocess t verge rte of er hour, d the verge time for ech cll is exoetil r.v. with me miutes. () ht is the robbility tht rrivig customer will fid the telehoe booth occuied? (b) It is the olicy of the telehoe comy to istll dditiol booths if customers wit verge of 3 or more miutes for the hoe. Fid the verge rrivl rte eeded to justify secod booth. () The telehoe service c be modeled s M/M/ queueig system with / & / /. The robbility tht rrivig customer will fid the telehoe occuied is P(L > 0), where L is the verge umber of customers i the system. Thus, P(L > 0) 0 ( - ) / 0.4 (b) q 3 ( ) 0. ( 0. ) from which we obti booth is 8 er hour. 0.3 er miute. Thus, the required verge rrivl rte to justify the secod

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