RF Egieerig otiuig Educatio Itroductio to Traffic Plaig Queuig Systems Figure. shows a schematic reresetatio of a queuig system. This reresetatio is a mathematical abstractio suitable for may differet arragemets i which users comete for a shared set of resources (servers). I everyday life, such arragemets are very commo, ad their aalysis rovides useful results with wide rage alicability. I this sectio we will address the queuig roblem i its geeral form. This aroach will allow us to treat various ractical roblems i traffic egieerig through a uified mathematical framework. Desite the geeral aroach, we will illustrate the uderlyig cocets by usig examles that are relevat to the field of cellular system traffic egieerig. Mea Arrival Rate Number of Users i the Queue Nq S S Iterarrival Time τ Queuig Time q S c Source Poulatio Geerated Traffic Queue Servers Figure.. A schematic reresetatio of a queuig system. Descritio of a Queuig System As evidet from Fig.., queuig systems are relatively comlex. Before we make a effort to aalyze them, we eed to defie some imortat terms ad variables.. Source oulatio (umber of subscribers). The source oulatio cosists of all users that are eligible for service i a give queuig system. I geeral, the most imortat roerty of the source oulatio is its size. From the stadoit of theoretical modelig, we make a distictio betwee fiite ad ifiite source oulatio. For the ifiite oulatio the average umber of service requests does ot deed o the umber of users that are curretly beig served. O the other had, for fiite oulatios, the robability of a ew service 9 Revisio.
RF Egieerig otiuig Educatio Itroductio to Traffic Plaig request decreases every time a user eters the queuig system. From a mathematical stadoit, the ifiite oulatio is easier to describe ad is frequetly used for traffic aalysis. I reality, every oulatio is fiite ad which oe of the two assumtios is used deeds o the ratio betwee the umber of otetial users ad the umber of available servers. If this ratio is large, we routiely assume that the oulatio is ifiite.. Arrival rate ad iterarrival time. The arrival rate is oe of the variables used to quatify the volume of geerated traffic. Withi the queuig system, the arrival rate is defied as a umber of service requests made i some secified time iterval. The ability of the queuig system to rovide effective service deeds ot oly o the mea arrival rate but also o its distributio. If the requests for service are evely saced i time, the queuig system ca rovide better service tha if the call attemts are clustered. As a illustratio, cosider two grahs showig the umber of call attemts for a imagiary cell site reseted i Fig... Both grahs have the same mea arrival rate of about call attemts er miute. However, the statistical behavior of the umber of call attemts i Fig.. (b) is much burstier. To assure that o calls are rejected, the umber of resources allocated to the site show i Fig.. (a) should be 3, while i Fig.. (b), we eed to allocate 4 resources. This is a sigificat differece (more tha 3%) ad it uderlies the imortace of the arrival rate distributio. umber of call attemts 5 45 4 35 3 5 5 5 umber of call attemts 5 45 4 35 3 5 5 5 3 4 5 6 time [mi] 3 4 5 6 time [mi] Figure.. Number of call attemts durig oe hour of cell site oeratio. Both grahs have average of call attemts er hour. The stadard way to secify the arrival rate is through distributio of iterarrival times. The iterarrival time is defied as the time iterval betwee two cosecutive service requests. The arrival rate ad iterarrival time are iversely roortioate. I other words, as the arrival rate icreases, the iterarrival time becomes smaller. 3. Servers. The server is a art of the queuig system caable of erformig a service task. The ractical imlemetatio of the server is determied by the tye of service that the queuig system is iteded to rovide. Examles of servers are: a comuter schedulig jobs that are set to a shared riter; a cashier i the suermarket, a toll booth o the highway ad so o. I cellular systems, the otios of the server ad the circuit are essetially the same. Table. secifies what ca be see as a server i various first ad secod geeratio cellular techologies. The art of the queuig system hostig servers is usually referred to as Revisio.
RF Egieerig otiuig Educatio Itroductio to Traffic Plaig the service facility. If all servers at the service facility are busy whe the call eters the system, the call must joi the queue ad wait for a server to become available. 4. Service time (all holdig time). The eriod of time over which a server is allocated to a idividual user is called the service time or the call holdig time. I geeral, the service time ca also be see as a radom variable. As i the case of the iterarrival times, erformace of the queuig system deeds fudametally o the service time distributio. For examle, i cellular etworks carryig redomiatly voice traffic, the exoetial distributio is commoly used to describe distributio of the service times. osider measuremets of the service time illustrated i Fig..3. The exoetial character of the distributio is evidet. The oly sigificat deviatio from the exoetial distributio occurs for brief service time duratio. The measuremets reseted i Fig..3 were collected i a cell servicig users with relatively low mobility. I cells where users are highly mobile, the distributio of holdig time deviates from exoetial for large call holdig time values as well. The reaso for deviatio resides i the hadoff rocess. Due to mobility, a user seds oly a ortio of the call holdig time withi the coverage area of a give cell. Therefore, the calls of extremely log duratio become highly ulikely. Histogram of call holdig time (HT), mea 9.6s, std 95.8sec, 498 measuremets.5 relative frequecy of occurace..5 Exoetial PDF, mea 9 sec 3 4 5 6 call duratio [sec] Figure.3. Histogram of the call holdig time measuremets For exoetial distributio of the call holdig time we ca write t Pr (.) { HT < t} ex Ts Revisio.
RF Egieerig otiuig Educatio Itroductio to Traffic Plaig where HT is the call holdig time ad T s is the distributio arameter referred to as the average call holdig time. The average call holdig time i cellular etworks varies as a fuctio of service rice, cultural differeces, time of the day ad umber of other arameters. Tyical values rage from to 8 secods. The quatity that is a iverse of the service time is the service rate. The service rate is defied as the umber of users that ca be rovided with the service i a give uit time rovided that the server is ever idle. For examle, for the distributio of the service times / E t / T. give i (.), the average service rate ca be calculated as [ ] s 5. Average resource occuacy traffic i erlags. The uit used i traffic egieerig as a measure for the server occuacy is called erlag (E). By defiitio, a sigle device occuied cotiuously or itermittetly for a total time t over some averagig time T carries traffic of t A [ E] (.) T From (.) we see that the maximum traffic that ca be carried by a sigle resource is E. The traffic of E corresods to the case whe the resource is occuied for the etire duratio of the averagig time iterval T. As a illustratio, cosider the grah i Fig..4. The grah secifies the occuacy of a server over some iterval T. It is imortat to ote that at ay give time, the resource is either occuied or ot. However, for a statioary eviromet, the average occuacy of the resource remais costat. t Average traffic t t 3 T t A t T t 3.5 8 4.5.5635 E 8 Figure.4. alculatio of the resource occuacy To assure a valid estimate of the average resource occuacy, the averagig time should be log eough. I cellular commuicatio, the tyical averagig time is hour. Sice the maximum traffic that ca be carried by a sigle resource has to be smaller tha, the total traffic carried by a service facility caot exceed the umber of resources. osiderig a grou of servers i Fig.., let t deote the sum of times durig which Revisio.
RF Egieerig otiuig Educatio Itroductio to Traffic Plaig exactly out of servers are held simultaeously withi the averagig eriod T. The total traffic carried by the grou ca be exressed as t t tc t A (.3) T T T T From (.3) we derive a differet iterretatio of the average traffic for multi-server systems. The exressio o the right had side of (.3) exresses the average umber of servers held simultaeously durig the averagig eriod T. This iterretatio allows easier measuremet of traffic carried by a grou of servers. The measuremet rocedure ivolves regular olig of the service facility ad loggig the umber of resources occuied at the measuremet time. 6. Offered, arried ad Lost Traffic. The average offered traffic is defied as A offered Ts (.4) T where is the average arrival rate, T s is the average call holdig time, ad T is the averagig eriod. For examle, if the rate of hoe call attemts at a give cellular site is calls/hour with a average call holdig time of 9 sec, the offered traffic is give as A offered T T s 9 36.5E (.5) Accordig to the alterative iterretatio for traffic i erlags, (.5) ca be see as the average umber of resources occuied at the service facility. Measuremet of the offered traffic requires cotiuous resource availability. I other words, every service request should fid a uoccuied resource ad be able to hold it for a desired eriod of time. Due to a relatively large variability i the offered traffic, this would require a large over-rovisioig of server resources. Although i some circumstaces it may be justified, the resource overrovisioig is ot regarded as a soud egieerig ractice. Most of the queuig systems are desiged to oerate with some robability that a articular service request will be deied. The robability of service deial is commoly referred to as the blockig robability. Figure.5 illustrates the resultig tradeoff i a case of a cellular system cell site. If the cell site is required to oerate with o blockig, the umber of assiged chaels eeds to be at least. However, it ca be see that with 8 assiged chaels, the ortio of time whe the cell site is blockig is oly mi durig the etire 6 mi of moitorig eriod. This ortio of time corresods to a blockig robability of /6.67%, which is assumed accetable i most cellular systems. Therefore, i ractice, oly a ortio of the offered traffic will be served. This ortio, referred to as the served traffic, ca be formally defied as A served t T (.6) 3 Revisio.
RF Egieerig otiuig Educatio Itroductio to Traffic Plaig where is the total umber of etwork resources, t is the eriod of time whe exactly resources are occuied, ad T is the time eriod used for date collectio ad averagig. Number of occuied chaels 8 6 4 8 6 4 Userved traffic Offered traffic Number of allocated chaels 3 4 5 6 time [mi] Figure.5. Relatiosh betwee offered, carried, ad lost traffic The differece betwee offered ad served traffic is commoly referred to as lost traffic. Real systems always oerate with a certai level of lost traffic. The task of the traffic laig egieer is to carefully balace the volume of the lost traffic agaist the umber of required resources ad rovide the most ecoomical solutio. 7. Service discilie (lost calls disositio). If at the time of service request arrival all resources are occuied, the request has to be laced i a queue. Whe oe of the resources becomes available, it will be allocated to oe of the requests i the queue. There are several differet algorithms used i determiig the order of the resource allocatio for the requests that are i the queue. These algorithms are commoly referred to as the queuig discilie. The most commo algorithm is the First ome First Serve (FFS), which is sometime referred to as the First I First Out (FIFO). I this algorithm the queuig system kees track of the order i which the requests are erformed, ad whe the resource becomes available, the same order is used for the resource allocatio. Examles of the FFS queuig discilie are a queue formed i frot of a airlie ticket couter ad a queue of ritig jobs i the rit server. Aother commo queuig discilie is the Last ome First Serve (LFS), which is sometimes referred to as the Last I First Out (LIFO). Accordig to this discilie the resources will be allocated i the oosite order of the order request arrivals. This queuig discilie accurately models behavior of the stack i comuter systems. Some other queuig discilies are ossible. I systems where the resource access is based o a versio of ALOHA rotocol, the queuig discilie is commoly referred to as the Radom Selectio Order (RSS) or the Service I Radom Order (SIRO). Accordig to this queuig 4 Revisio.
RF Egieerig otiuig Educatio Itroductio to Traffic Plaig discilie, each service request i the queue has the same robability of beig selected for service oce oe of the resources i the service facility becomes available. May queuig systems aly some sort of riority queuig i which the system erforms the resource allocatio o a basis of request riority. For examle, i commuicatio systems that are desiged to suort simultaeous voice ad data commuicatio, voice traffic routiely receives higher riority tha data. The queuig discilie has a sigificat imact o the erformace of the queuig system. Parameters like the average delay time, the average umber of users i the queue, the robability of excessive delay, ad the robability of the user deflectig from the queue all deed o the eforced queuig discilie. For that reaso, whe a give queuig system is aalyzed, the queuig discilie eeds to be take ito accout. 8. Maximum Queue aacity. Oe of the mai characteristics of the queuig system is the caacity of its queue. The caacity of the queue is defied as the umber of service requests that it ca hold. Based o the queue caacity, systems ca be classified as either lossless or lossy. I lossless systems, the caacity of the queue is ifiite ad every service request is allowed to wait util a resource becomes available. I lossy systems, the queue has a limited caacity ad oly a limited umber of user requests ca be laced i the queue. If the umber of requests exceeds the queue caacity, the request is deied or blocked. A extreme case of the lossy queuig system is the system with queue caacity equal to zero. This system is commoly referred to as the loss system. Deedig o the goals of traffic egieerig the queuig system i Fig.. is aalyzed for differet asects of its erformace. Examles of some relevat erformace measures that would result from such aalysis are give as []: Exected umber of the service request i the queuig system Exected umber of requests i the queue Traffic carried by the servers Lost traffic Probability of request blockig Average waitig time Average time sed i the queuig system, Server utilizatio alculatio of each of the above erformace measures is ot a trivial task sice it requires a thorough queuig system descritio. I geeral, some assumtios eed to be made regardig the behavior of the user oulatio, ad distributio of the iterarrival ad service times. The accuracy of the assumtios will limit the accuracy of the mathematical model ad hece, the accuracy of the obtaied results. Sice the erformace of the queue chages drastically as a fuctio of adoted assumtios, aalysis of a geeral queuig system is a challegig task. For that reaso the queuig systems are divided ito several classes ad the aalysis of each class is erformed ideedetly. A method for the queuig system classificatio will be described i sectio.4. 5 Revisio.
RF Egieerig otiuig Educatio Itroductio to Traffic Plaig. Poisso Process of Radom Arrivals As reviously discussed oe of the most imortat assumtios regardig the queuig system is the distributio of the service request iterarrival times. The iterarrival times are roerty of the user oulatio, ad i geeral, they deed o may factors. For examle, i cellular systems the call origiatio rocess is a fuctio of the habits of mobile hoe users, their lifestyle, occuatio, mobility atter ad so o. A similar situatio arises i other queuig systems as well. However, extesive observatio ad measuremets have revealed that i may systems the service requests assume behavior of a Poisso rocess. Havig i mid large variability betwee differet queuig systems, this is a remarkable result. For that reaso, i this sectio we rovide a brief descritio of the Poisso rocess. May ractical methods used i cellular system traffic egieerig that are reseted i the subsequet sectios will be based o the assumtio of Poisso service request arrivals. osider a stochastic rocess that rovides a cout of a certai radom evet i a give time iterval startig from some coveietly chose origi. Let this rocess be described as a fuctio of time N ( t). For ay articular realizatio the fuctio N ( t) will be a "staircase" like fuctio gradually steig through the ositive itegers. A rocess of such ature is commoly referred to as the coutig rocess ad it ca be formally defied as follows []: Defiitio.. A stochastic rocess ( t) coditios are satisfied:. N ( ). N ( t) assumes oly oegative iteger values 3. t < t imlies that N( t ) N( t ), i.e. ( t) 4. ( t ) N( t ) t, that is i the iterval ( t ]. N costitutes a coutig rocess if the followig N is o-decreasig iteger fuctio, ad N is the umber of radom evets that have occurred after t but ot later tha, t A examle of the coutig rocess realizatio is show i Fig..6. From the coditios that are give i Defiitio., ad the grahical reresetatio i Fig..6, we see that the coutig rocess ca be used to model the service request arrivals i a queuig system. I other words, the grah of the fuctio N ( t) show i Fig..6 may be see as a cout of the umber of service, t. requests that have arrived i the time iterval ( ] 6 Revisio.
RF Egieerig otiuig Educatio Itroductio to Traffic Plaig N( t) 5 iterarrival time time t time Occurrece of radom evets Figure.6. Examle of a coutig rocess realizatio t The Poisso rocess is a coutig rocess that satisfies some additioal requiremets. These requiremets are give as:. For every two o-overlaig time itervals ( t, t ] ad (, t 3 4] are ideedet radom variables. I other words ( t ) N( t ) ( t ) N( ) t the umber of the evets N is ideedet from N 4 t 3. Therefore, the Poisso rocess is a coutig rocess with ideedet icremets.. Distributio of evets i ay give iterval deeds oly o the legth of the iterval ad is ideedet from the actual time of its begiig. I other words, the Poisso rocess has statioary icremets. 3. The robability that exactly oe evet occurs i a time iterval of legth h is give by { N( h) } h o( h) P, where is a costat. 4. The robability that more tha oe evet occurs withi the time iterval of duratio h is give by { N( h) > } o( h) P 7 Revisio.
RF Egieerig otiuig Educatio Itroductio to Traffic Plaig I the requiremets 3 ad 4, symbol o ( h) idicates a fuctio that teds towards zero faster tha h itself. I other words, as h becomes smaller, the effects of o ( h) ca be eglected. I summary, i a Poisso rocess, the evets occur oe-at-the-time ad at a costat rate equal to. I additio, the rocess stays ideedet of the begiig of the observatio time. Fially, the Poisso rocess "has o memory". The distributio of evets i a give iterval does ot deed o the distributio i ay revious o-overlaig iterval, or will it imact the distributio of evets i ay future o-overlaig iterval. There are several imortat roerties of the Poisso rocess that ca be derived from its defiitio. The two most imortat oes are give as follows. Proerty. Let ( t) N be a Poisso rocess with the arameter. The radom variable describig the umber of evets i ay give iterval of legth t is give as Proerty. Let ( t) t P Y (.7) { k} ex( t) ( ) k! k N be a Poisso rocess with the arameter. The iterarrival time betwee evets is a exoetially distributed radom variable with mea give as /. I other words the robability desity fuctio of the iterarrival times is give as ( τ ) ex( τ ) df (.8) Proof of the above two roerties ca be foud i [-3]. Here, we rovide some examles that will illustrate the use of (.7) ad (.8). Examle.. osider a Poisso rocess with the arameter. alculate the average time betwee two cosecutive evets ad the average rate of the evet occurrece. The average time betwee evets ca be calculated as τ E { τ} τ ex( τ ) dτ The average rate is give as r / E{ τ }. Therefore, the distributio arameter ca be iterreted as the average rate of the evet arrivals. Examle.. Assume that the umber of call arrivals i a give cell of a cellular system may be modeled as a Poisso rocess with a average rate of calls er miute. What is the average iterarrival time? What is the robability of receivig more tha 5 calls er miute? Usig the results of the revious examle, we have 8 Revisio.
RF Egieerig otiuig Educatio Itroductio to Traffic Plaig τ /6 6 [ sec] The robability of receivig more tha 5 calls ca be foud usig Pr Usig (.7), we have Pr { Y > 5} Pr{ Y 6} Pr{ Y 7} { Y > 5} Pr{ Y 5} Substitutig the umerical values Pr { Y > 5} 5 k ( ) k! k ex 5 k ( ) ( t) k! k ex.487 ( t) As ca be see, although the average umber of calls er miute is, about 5% of the time, the actual umber of calls laced withi oe miute will be more tha 5. Therefore, to assure that most of the calls are served, the umber of chaels at the site has to be larger that. Examle.3. osider the measuremets i Table.. The measuremets reort the umber of jobs set to a riter server o a miute by miute basis for a eriod of oe hour. Determie if the rocess ca be modeled as the Poisso rocess ad if that is the case, estimate the average rate of service request arrivals. Table.. Measuremets reortig the umber of jobs serviced by a riter server time # jobs time # jobs time # jobs time # jobs time # jobs 6 3 5 37 6 49 9 6 4 5 6 8 38 5 5 4 3 7 5 8 7 5 39 6 5 4 5 6 5 8 4 4 5 5 5 7 9 3 4 5 53 3 6 6 8 7 3 4 8 54 7 7 9 9 3 3 6 43 3 55 5 8 6 3 4 44 7 56 9 7 6 33 4 45 57 4 5 6 34 3 46 8 58 7 3 5 35 4 47 59 6 5 4 4 36 5 48 6 4 9 Revisio.
RF Egieerig otiuig Educatio Itroductio to Traffic Plaig First we will estimate the average rate of job arrival 6 ( 6 6 7 5 6 4) 5. 833 arrivals/mi Usig Table., we ca calculate the ormalized frequecy of occurrece which ca be used as a estimate of the robability mass fuctio of a discrete rocess. The ormalized frequecy of occurrece is calculated i accordace with: Normalized Frequecy Number of Occurreces 6 For examle, the ormalized frequecy of occurrece for three jobs withi a miute is give by F 5 6 3.833 Figure.7 shows the lot of the relative frequecy of occurrece derived from the measuremets i Table.. O the same lot we show the values for the robability mass fuctio of a ideal Poisso rocess that has the same mea rate of arrivals. As evidet, the differece is relatively small ad for ractical traffic dimesioig of this system we may assume that the rocess of service request arrivals is a Poisso rocess. Frequecy of occurece / robability..8.6.4...8.6.4. 3 4 5 6 7 8 9 Number of jobs er miute Figure.7. omariso of the frequecy of occurrece lot ad the PMF of the ideal Poisso rocess for data i Examle.3 Revisio.
.3 Birth ad Death Processes RF Egieerig otiuig Educatio Itroductio to Traffic Plaig I the revious sectio we cosidered the Poisso rocess ad saw that it ca be used to describe the arrivals of service requests i may cases of great ractical iterest. I a ractical queuig system, the request arrivals result i resource allocatio ad evetually the users get served ad leave the queue. It is customary to view this rocess as a member of a wider class of stochastic rocesses that are commoly referred to as the birth ad death. Withi this framework, every icomig request is regarded as a birth ad every user that, after beig served, leaves the system is regarded as a death. For the Poisso rocess the average birth rate is secified by the distributio arameter. The birth rate ca chage as a fuctio of the state of the queuig system. However, we ca still say that i a short time iterval h, the robability of a sigle birth is equal to h o( h), where subscrit idicates oe of the system states. Likewise, it is reasoable to assume that i a short time iterval h, the umber of users leavig the system is equal to h o( h), where idicates the average death rate, ad idex refereces the state of the queuig system. The birth ad death rocess is frequetly used as a mathematical model of a queuig system ad i this sectio we rovide its descritio. The framework of the birth ad death rocess will allow us to derive some results that describe the behavior of the queuig systems i geeral. The formal defiitio of the birth ad death rocess is give as []: Defiitio.. osider a stochastic rocess N ( t) that is cotiuous i time but has a discrete state sace Ω {,,, }. Suose that this rocess describes a hysical system that is i state E,,,, at time t, if ad oly if N ( t). The the system is described by the birth-addeath rocess if there exist oegative birth rates,,,,, ad oegative death rates,,,,, such that the followig ostulates (sometimes called earest eighbor assumtios) are true:. State chages are oly allowed betwee state E to state E or from state E to E if, but from state E to state E oly.. If at time t the system is i state E, the robability that betwee time t ad time th a trasitio from state E to state E occurs equals h o( h), ad the robability of trasitio from E to E is h o( h) (if ). o h. 3. The robability that i time iterval from t to th more tha oe trasitio occurs is ( ) Before we roceed, let us rovide some examles of rocesses that ca be classified as birth ad death. As a first examle, let us cosider a stochastic rocess modelig the oulatio i a closed system. The assumtio that the system is closed is ecessary to assure that the oly mechaisms that the oulatio ca chage are death ad birth. Referrig back to Defiitio we ca idetify the followig aalogies: Revisio.
RF Egieerig otiuig Educatio Itroductio to Traffic Plaig. State of the system E reresets the total oulatio cout at a give time t. Oe should ote that although we ca determie the oulatio at a arbitrary time (that is the rocess is cotiuous i time), the actual values of E ca be oly oegative itegers, i.e.,,, etc. Therefore, the rocess is cotiuous i time, but desecrate i state sace.. At ay give state E, the oulatio icreases at the rate of ad decreases at the rate of. Obviously, the oulatio exerieces growth if > ad it is subject to declie if >. I actual hysical systems, o deaths or births ca occur if the system is i state E. However, defiitio of the birth-ad-death rocess allows for a ozero birth rate eve whe the system is i state E. 3. Let s assume that we cout the oulatio at times t ad th. If the time icremet h is ket small we exect the robability of birth to be give as the roduct of birth rate ad the give time icremet, i.e.,. Similarly, the robability of death is give as h. Give that h h is ifiitesimally small, the robability of both death ad birth occurrig withi such a small time icremet ca be eglected, that is assumed as essetially zero. 4. As a fial ote, we oit out that i geeral, birth ad death rates are a fuctio of the curret oulatio cout. I other words, if the oulatio grows, both the rate of birth ad the rate of death ca be exected to grow. Likewise if the oulatio lummets, the rate of birth ad the rate of death decrease as well. However, if the oulatio is very large, the imact of the actual oulatio cout o the birth ad death rates becomes smaller. I a boudary case for ifiite oulatio we would exect the rates of death ad birth to remai costat. As a secod examle, let us examie the modelig of traffic served i a cell of a cellular commuicatio system.. The state of system E reresets the total umber of users that are beig served by a give cell. Ulike the revious examle i which the set of ossible states ecomasses all ositive itegers, the ossible states i this case are limited by the umber of available resources at the cell site. I other words, E {,,,,}, where is the umber of truks (that is, voice chaels) that are available at the site.. The rocess of birth is aalogous to a ew user tryig to set u a call. Therefore, the birth rate gives the rate at which the users request the service. I a similar way the death corresods to a user that has comleted the call ad released the voice chael..3. State Diagram Reresetatio of Birth ad Death Process A useful visualizatio of the birth ad death rocess is rovided through the state trasitio rate diagram. A examle of such a diagram is give i Fig..8. Revisio.
RF Egieerig otiuig Educatio Itroductio to Traffic Plaig State: - 3 Figure.8. A examle of the state diagram for birth ad death rocess The umber iside the circle idicates the state of the system. For examle, i a cellular system this would be the umber of users serviced by a give site. Values idicate the birth rates at each of the system states. Similarly, values i rereset the death rates. The state diagram allows oly the earest eighbor trasitios ad oly the birth trasitio is allowed from state zero. State diagram reresetatio of the birth ad death rocess will be frequetly used for aalyses reseted i subsequet sectios. For that reaso, we derive differetial-differece equatios for P ( t) Pr { N( t) }, that is, the robability that the system is i state E at time t. Note that the derivatio reseted here is geeralized, ad as such, it is valid for ay system that ca be described usig the birth ad death rocesses. If P t h that at the time t h system will be i the state E has four comoets listed as follows:, the robability ( ). The system was i state E at time t ad o births or deaths have occurred. Kowig that the robability of birth is h o( h) ad the robability of death is h o( h), this comoet ca be exressed as: i P ( ) ( t h) P ( t) [ h o( h) ][ h o( h) ] P ( t)( h h) o( h) (.9). The system was i state E at time t ad a birth has occurred. The robability of this evet is give as: P ( ) ( t h) P ( t) h o( h) (.) 3. The system was i state E ad a death has occurred. The robability of this evet is give as: P ( 3 ) ( t h) P ( t) h o( h) (.) 4. Two or more trasitios have occurred. By the roerties of the birth ad death rocess stated i Defiitio, this robability is: 3 Revisio.
RF Egieerig otiuig Educatio Itroductio to Traffic Plaig ( 4 ) ( t h) o( h) P (.) From (.9) through (.) we have: or P 4 i P ( i) ( t h) P [ h h] P ( t) hp ( t) hp ( t) o( h) ( t ) P ( t) h By lettig h, (.4) reduces to: dp dt ( t) (.3) ( h) o ( ) P ( t) P ( t) P ( t) (.4) h ( ) P ( t) P ( t) P (.5) Equatio (.5) is valid for. For, followig the some rocedure oe obtais: dp dt ( t) ( t) P ( t) P (.6) If the iitial state of the system is P ( ), ad ( ) i j E i, the iitial coditios are give as: P, for j i (.7) From (.5) ad (.6), we see that the birth ad death rocess ca be described usig a ifiite set of differetial equatios, with iitial coditios give i (.7). Although it ca be rove that the solutio of these equatios exists uder very geeral circumstaces [], it ca be rarely obtaied i a aalytical form. The steady state solutio of (.6) ad (.7) are of a secial ractical iterest. The steady state solutio assumes that a sufficiet time has elased ad that the system has reached statistical equilibrium. I a steady state, all system state robabilities ( P ( t) values), become costat ad hece the derivatives o the left-had sides of (.5) ad (.6) are equal to zero. Therefore, uder the steady state assumtios ad ( ), for (.8), for (.9) Equatio (.9) ca be rewritte as 4 Revisio.
RF Egieerig otiuig Educatio Itroductio to Traffic Plaig Revisio. 5 (.) Also, (.8) ca be rearraged i the form (.) Sice (.) is valid for every, usig (.) we ca coclude that for,,, (.) Usig (.) we ca comute (.3) (.4) 3 3 3 3 (.5) I geeral, we have (.6) Sice the sum of all state robabilities has to be equal to, S (.7) Fially, as a summary, we have ( ) { } S t N P r, (.8) ad ( ) { } S t N P r (.9) where, ad S (.3)
RF Egieerig otiuig Educatio Itroductio to Traffic Plaig From (.8) through (.3) we see that the birth ad death rocess has a steady state solutio if the sum S coverges. I such a case, there is a fiite robability of a system occuyig state zero. This would mea that from time to time the system catches u ad maages to serve all users. O the other had, if S diverges, this idicates of a ustable system i which births are occurrig at faster rates tha deaths. For ractical alicatios of the birth ad death rocesses, we will assume that the system is ot ustable, that a steady state exists, ad that the state robabilities are costat ad give by (.9)..3. Little's Formula Little's formula is a simle but very imortat equatio that alies to ay system i equilibrium i which customers arrive, sed some time ad the deart. The formula is give by L W (.3) where L is the average umber of customers i the system, is the average rate of customer arrivals, ad W is the average time that customers sed i the system. The roof of (.3) is relatively comlex ad is beyod the scoe of this documet. To get a ituitive uderstadig of Little's formula, cosider a system with a sigle server ad a ifiite queue. If the average service time is W, the umber of users that arrive while oe user is beig served is W. Sice the resource is occuied, these users are laced i queue ad the state of the system is described by (.3). The most imortat asect of (.3) is its uiversal alicability, therefore it is used frequetly throughout this documet. Examle.4. As a illustratio of a birth ad death rocess, cosider a queuig system havig oly oe server. Assume that that the service request arrivals ca be accurately modeled as a Poisso rocess with a average rate of mi, ad that the average time required to service oe request is give by W s.5 mi. Also assume a ifiite queue caacity with a FIFO queuig discilie. This kid of queuig system ca be used to model may ractical "real life" scearios. For examle, it ca be used to model the queue formed at the riter server, or the queue formed i a suermarket with oly oe cash register. Estimate the robability that exactly users are i the queue, a average umber of users i the queuig system, ad the average time that users sed i this queuig system. First, we estimate the average death rate, that is, the average rate at which the users would be leavig the system rovidig that the server has o idle time. This rate is estimated as: mi (.3) W s.5 Usig (.9) ad (.3) we have ad (.33) 6 Revisio.
RF Egieerig otiuig Educatio Itroductio to Traffic Plaig S (.34) 3 Therefore, the robability of havig exactly users withi the queue is give by: (.35) S The average umber of users i the system ca be calculated as L (.36) 4 8 Multilyig both sides i (.36) with we obtai L ( ) (.37) 4 8 6 Subtractig (.37) from (.38) L 4 8 (.38) 4 4 Therefore the average umber of users i the queuig system is give by L The average time that users sed i the system ca be calculated usig Little's formula as L W mi (.39).4 Kedall's Notatio Kedall's otatio is frequetly used for describig queuig systems of various roerties. This is a shorthad otatio i the followig form A/B//K/m/Z where the iterretatio of idividual terms is as follows: A B K - distributio of the iterarrival times - distributio of the service times - umber of servers withi the service facility - maximum umber of users withi the queuig system 7 Revisio.
RF Egieerig otiuig Educatio Itroductio to Traffic Plaig m Z - size of the user oulatio - service discilie Withi Kedall's otatio for the descritio of the arrival rocess ad service times, the followig symbols are used: GI G H k E k M D U - geeral ideedet arrival/service times - geeral (ot ecessarily ideedet) arrival/service times - k-stage hyerexoetial distributio - Erlag-k distributio - exoetial distributio (Poisso rocess) - costat iterarrival/service times - uiform distributio As a illustratio, cosider the queuig system described i Examle.4. I Kedall's otatio, this queue ca be described as follows. Sice the arrivals are modeled usig the Poisso rocess A M. Due to exoetially distributed service times B M. Sice there is oly oe server,. Both the queue ad the oulatio are of a ifiite size ad therefore K ad m. As the queuig discilie is First-I-First Out, Z FIFO. Therefore, Kedall's otatio for the queuig system i Examle.4 is M/M// / /FIFO. Very ofte, if the queue ad oulatio are ifiite ad the queuig service discilie is FIFO, the last three desigators of the otatio are omitted. I this examle, the otatio would reduce to M/M/..5 Examles I this sectio we illustrate the alicatio of the queuig theory i the aalysis of some commoly ecoutered queuig systems. Two examles will be reseted. The first examle aalyzes the roblem of coectig two workstatios to a cetral server. The secod examle shows the alicability of the queuig theory i the desig of reliable microwave commuicatio liks. Examle.5. osider a roblem illustrated i Fig..9. Two work statios eed to be coected to a sigle server ad we examie two ossible cofiguratios that ca be used to accomlish the task. I the first cofiguratio, the coectio is achieved by usig two searate lies. The secod cofiguratio uses oe lie with a badwidth that is two times larger. Let us assume that each workstatio geerates messages er secod ad that the average for the message delivery is give as for the idividual lies ad ( ) for the lie with the larger badwidth. Both cofiguratios i Fig..9 ca be modeled usig the theory develoed i revious sectios. We will examie some erformace matrix as they are observed from idividual workstatios. 8 Revisio.
RF Egieerig otiuig Educatio Itroductio to Traffic Plaig Work statios ofiguratio Server ofiguratio Figure.9. Two differet cofiguratios examied i Examle.5 ofiguratio. I cofiguratio, we essetially have two searate M/M/ queuig systems with the same erformace. Usig the results of the birth ad death rocess aalysis (c.f. Sectio.3), the robability of havig exactly messages i a trasmissio lie (or associated buffer), is give by where ad S, (.4), ρ ρ (.4) 3 S (.4) ρ Therefore, ( ρ ) ρ (.43) The average umber of messages withi each of the trasmissio lies is give by ρ ρ (.44) ρ ( ) ρ ( ρ ) ρ ( ρ ) ρ Usig Little's formula, the average time required for the message delivery is give by 9 Revisio.
RF Egieerig otiuig Educatio Itroductio to Traffic Plaig ρ W (.45) ( ρ ) Therefore, i the first cofiguratio each of the workstatios exerieces a average throughut R (.46) W ofiguratio. ofiguratio ca be see as oe M/M/ queuig system with a birth rate of ad a death rate of. Followig the same aroach as i the case of cofiguratio, we obtai the followig results ad ( ρ ) ρ (.47) ρ (.48) ρ W (.49) ( ) ( ) R (.5) ( ) R W Therefore, the secod cofiguratio is two times more efficiet tha the first oe. Examle.6. I this examle we illustrate the imact of the lik diversity o the reliability of a microwave coectio. osider a microwave lik with a hot stadby [4]. Let us assume that a mea time betwee a sigle lik failure is give as T. Whe a lik fails (either the mai oe or the hot stadby), the mea time to reair is give by T r. If we assume the same reliability of the mai lik ad the hot stadby, let us estimate the reliability imrovemet over a system without the lik diversity. The microwave lik i this examle ca be modeled as a birth ad death rocess with just three states ad the state diagram show i Fig... Figure.. State diagram for the microwave system i Examle.6 f 3 Revisio.
RF Egieerig otiuig Educatio Itroductio to Traffic Plaig The state of the system corresods to the umber of o-workig liks. I other words, state corresods to the case whe both the mai lik ad its hot stadby are oeratioal; state corresods to the case whe oe of the liks fails; ad state corresods to failure of both the mai lik ad the hot stadby. The birth ad death rates are idicated i Fig.., where ad, (.5) T f (.5) T r To calculate the mea time betwee the failure for the system with the lik diversity we use the diagram i Fig.. to estimate the steady state rate at which the system reaches state. From Fig.., this rate ca be calculated as (.53) f where (.54) S Therefore, f (.55) ad the time betwee the failures becomes T f f Tr T T T r f f (.56) To illustrate the resultig imrovemet, let us cosider the followig umerical data. The average time betwee lik failure is T 4 hours ad the average reair time is f T r 4 hours. Whe the lik diversity is used, the average time betwee failures becomes 4 T 4 f 337,333 [hours] (.57) 4 4 which is a sigificat imrovemet. 3 Revisio.