Effect of walking-distance on a queuing system of totally asymmetric simple exclusion process equipped with functions of site assignments
EE ff ect of walking-distance on a queuing system of totally asymmetric simple exclusionprocess equipped with functions of site assignments Satori Tsuzuki a , Daichi Yanagisawa a , Katsuhiro Nishinari a a Research Center for Advanced Science and Technology, The University of Tokyo, 4-6-1, Komaba, Meguro-ku, Tokyo 153-8904, Japan
Abstract
This paper proposes a totally asymmetric simple exclusion process on a traveling lane, which is equipped with a queueing systemand functions of site assignments along the parking lane. In the proposed system, new particles arrive at the rear of the queueexisting at the leftmost site of the system. A particle at the head of the queue selects one of the empty sites in the parking laneand reserves it for stopping at once during its travel. The arriving time and staying time in the parking sites follow half-normaldistributions. The random selections of empty sites are controlled by the bias of the exponential function. Our simulation resultsshow the unique shape of site usage distributions. In addition, the number of reserved sites is found to increase with an S-shapecurve as the bias to the rightmost site increases. To describe this phenomena, we propose an approximation model, which is derivedfrom the birth-death process and extreme order statistics. A queueing model that takes the e ff ect of distance from the leftmost siteof the traveling lane into consideration is further proposed. Our approximation model properly describes the distributions of siteusage, and the proposed queueing model shows good agreement with the simulation results. Key words:
Totally Asymmetric Simple Exclusion Process, Queueing Model, Statistical Mechanics, Assignment Problems,Multi-Particle Physics
1. Introduction
The queueing theory, which was started by A. K. Erlang[1]at the beginning of 20th century, has attracted many scientistsand researchers. Most of the theory is still in veil; neverthe-less, a strong demand for this theory exists not only in aca-demic studies of non-equilibrium statistical physics but alsoin many engineering fields such as tra ffi c system[2], humandynamics[3, 4], and molecular motor transport[5, 6]. The studyof queueing systems has been associated with the totally asym-metric simple exclusion process (TASEP) because of two mainfeatures: transportation in a one-way direction and volume ex-clusion e ff ect, which are suitable for the simulation of queueingsystems[7, 8, 9].This paper proposes a totally asymmetric simple exclusionprocess on a traveling lane, which is equipped with a queue-ing system and site assignments along the parking lane, underopen boundary conditions. In the proposed system, new parti-cles arrive at the rear of the queue existing at the leftmost siteof the system. Thereafter, a particle at the head of the queueselects one of the empty sites in the parking lane and reserves itfor stopping once during its travel. The arriving time and stay-ing time in the parking sites follow half-normal distributions.The random selections of empty sites in the site assignmentsare controlled by the bias of the exponential function. Email addresses: [email protected] (SatoriTsuzuki), [email protected] (Daichi Yanagisawa), [email protected] (Katsuhiro Nishinari)
Similar mechanics of the proposed system can be observed inmany real-world cases (e.g. parking problems in highways, air-plane boarding, and airport ground transportations). Therefore,studying the proposed system is meaningful in many applica-tion fields. In particular, it is important to investigate the rela-tionship between the occupancy of parking sites and the waysof site assignments because they are closely related with eachother. The major scope of this research is to describe the rela-tionship between the e ff ect of site assignments in the proposedsystem and the occupancy of parking sites.Because the parking site can be regarded as an absorption sitein a wider sense, the proposed model is classified into the samecategory of the studies on multiple-lane systems with LangmuirKinetics. Many previous studies have been reported, exempli-fied by [10, 11] for parallel-lane systems under periodic con-ditions, [12] for triple parallel-lane systems with Langmuir Ki-netics, and [11, 12, 13, 14, 15, 16, 17] for two parallel-lane sys-tems with Langmuir Kinetics. However, queueing problems arenot taken into accounts in these studies. Regarding the functionof site assignments, our previous research[18] is a pioneeringstudy on site-assingment for parallel-lane systems; however, theproblem of queueing was not discussed in that study.In the modeling of queues, we consider the e ff ect of walk-ing distance from the entrance. Several previous studies haveworked on this problem of walking-distance for specific cases,exemplified by the D-Fork system[19] , D-Parallel system[19],and combinational queueing system of D-Fork with D-Parallelsystem[20]. A distinguishing factor of the proposed systemcompared to these previous systems is that the parking sites Preprint submitted to Phys. Rev.E July 24, 2018 a r X i v : . [ n li n . C G ] J u l igure 1: Schematic view of the target system. (service sites) push the particle back to the traveling lane,whereas in the previous studies, the particles pass through theservice sites and exit from the opposite side of the travelinglane. Since the particle reentering the traveling lane causes de-lay in the traveling lane, the ways of site usage distributionsa ff ect the queueing system. Hence, the mechanics in the pro-posed system are di ff erent from those of the previous studies.In this paper, we propose an approximation model to describethe site usage distribution of the proposed system on the basis ofbirth-death process for the spatial direction and extreme orderstatistics for the time direction.The remainder of this paper is structured as follows. Sec-tion 2 provides a summary of the target system and that of theclassical M / M / c queueing model. Section 3 investigates the de-pendence of the utilization of parking sites on the distributionparameter of the exponential function through simulations. InSection 4, we propose an approximation model that describesthe site usage distribution of the proposed system on the ba-sis of the birth-death process and extreme statistics. Section 5summarizes our results and concludes this paper.
2. Models
A schematic view of the target system is depicted in Fig.1.The system consists of two parallel lanes: a traveling lane,which is composed of L sites, and a parking lane, which iscomposed of N s sites ( N s = L / ∆ l is set to be 2. In the proposed system, a parti-cle takes four di ff erent states. The green state indicates that theparticle is in a state of queuing. The red state indicates that theparticle is in transport before stopping at the designated site ofthe parking lane. The yellow state indicates that the particle iscurrently stopping at the site of the parking lane. The blue stateindicates that the particle is in transport after exiting the parkinglane. To sum up, the flow of a particle is described as follows.A particle arrives at the rear of the queue, which emerges atthe leftmost site. After staying in the parking site, the parti-cle goes back to the traveling lane, changing its state from theyellow-colored state to the blue-colored state, and then moves towards the rightmost site. The particle in the traveling lane iseliminated from the system at the next step after moving to therightmost site.Additionally, a parking site takes three kinds of states (re-served state, occupied state, and empty state). We call boththe occupied state and reserved state simply as “a busy state” inthis study. For the time integration, we adopt the parallel updatemethod.The interesting subjects of this study are not only the prob-lems with the random arrival such as the parking in highwaysbut also the problems with the scheduled arrival with the ran-dom delay such as the airport ground transportations. Espe-cially in the latter case, the use of normal distribution is rea-sonable, however, it has a practical problem in that we have tocut the tail of the left side of the distribution in some cases. Inorder to avoid this problem, we take up to use the half-normaldistribution in this study.The interval of the arrival time is set to follow a half-normaldistribution; the mean τ in and deviation σ in of the interval ofarrival time are given as follows: τ in = ¯ τ in + ¯ σ in (cid:114) π (1) σ in = ¯ σ in (cid:114) (1 − π ) (2)Here, ¯ τ in and ¯ σ in are the mean and deviation of the originalnormal distribution, respectively. A schematic view of τ in , σ in ,¯ τ in and ¯ σ in is depicted in Fig.2.A particle at the head of the queue selects one of the emptysites in the parking lane and reserves it for stopping once duringits travel. Similar to that of the arrival time, the interval of thestaying time is also set to follow a half-normal distribution: τ s = ¯ τ s + ¯ σ s (cid:114) π (3) σ s = ¯ σ s (cid:114) (1 − π ) (4)Here, τ s and σ s are the mean and deviation of the half-normaldistribution, while ¯ τ s and ¯ σ s are those of the original normaldistribution, respectively. Unless otherwise noted, the mean anddeviation of the two cases are indicated by τ in , σ in , τ s , and σ s in this paper.Two important rules are made to the system. The particle atthe head of the queue is not permitted to reserve the parkingsite that has already been reserved by the other particle unlessthe site releases the particle. In addition, the yellow particlesin the parking sites have priority access to the upper site on thetraveling lane compared to the red / blue colored particle, whichis to access the same traveling site.The random selection of an empty parking site by the par-ticle at the head of the queue is controlled by the exponentialfunction ke − kx ( x ≥ , k > igure 2: Schematic view of the parameters of the interval of arrival time whichfollows half-normal distribution. The horizontal axis indicates the time step andthe vertical axis indicates the probability. toward the opposite rightmost site by setting the reversed se-quential number to the number of sites L . Note that the inversetransform sampling (ITS) [21, 22] is introduced to generate ran-dom variables that follow the exponential distribution.We denote the latter type of bias by multiplying the negativesign to parameter k for the sake of easy view. Namely, in thenotation of ke − kx , the random selection gets biased towards theleftmost site as parameter k increases while k >
0. In contrast, itgets biased towards the rightmost site as parameter k decreaseswhile k <
0. In the case of setting parameter k to be zero, noexternal bias is given to the random selection. / M / c queue In this section, we overview the classical queueing theory.A queueing system is characterized by six stochastic proper-ties: the arrival process A , service process B , number of serversin the system C , maximum number of possible customers whowill arrive at the system K , number of sources of customers N ,and service discipline D . All these properties are summarizedas A / B / C / K / N / D by Kendall’s notation[23]. The notation of K and N are abbreviated in case of infinity and that of D is abbre-viated in case of FIFS (First Come First Served); in this case,the system can be represented simply as A / B / C . The proposedsystem in this study is categorized into M / M / c queueing sytemsbecause the arriving time and staying time follows Markov pro-cess and the system has finite number of parking sites N s . Notethat the left and right M indicate the Markov process, and thenotation c indicates the number of servers (the c corresponds to N s in the system). In this section, several important formulasof the classical M / M / c queueing system are enumerated. Formore details on queueing theories, refer to[24, 25]The arrival rate λ and service rate µ are defined as the char-acteristic values of the queueing system. On the condition thatthe λ and µ are given as constant parameters, a distribution ofprobability P n that the whole system (including queue) has n customers at a stationary state is obtained as a consequence ofsolving the transition equation of length of queue between thetime step n and time step n + P n = a n n ! P ( n = , , . . . , c ) a n c n − c c ! P ( n = c + , c + , . . . ) (5) P = (cid:40) c − (cid:88) n = a n n ! + a c c ! 11 − ρ (cid:41) − (6) a = λµ (7) ρ = λ c µ (8)Additionally, the length of queue L q , total number of customersin the whole system L , and the number of utilized servers U areobtained from Eq.(5) to Eq.(8), as follows: L q = ∞ (cid:88) n = c + ( n − c ) P n = C ( c , a ) ac − a (9) C ( c , a ) = cc − a a c c ! P (10) L = ∞ (cid:88) n = nP n = L q + a (11) U = L − L q = a (12)Consequently, the utilization of servers corresponds to the pa-rameter a , which is defined as the value of arrival rate dividedby service rate, as shown in Eq.(7).In the case of c = M / M / λ andservice rate µ are defined as the inversed values of arrival timeand service time, as follows: λ = τ in − (13) µ = τ s − (14)Even in general cases of c >
1, all the servers are assumed tohave same the values of arrival rate λ and service rate µ definedby Eq.(13) and Eq.(14), respectively, similar to the M / M / M / M / c queueing model. However, thisassumption causes a non-negligible deviation from real-worldsystems because the e ff ect of walking distance to each serveris not considered in the classical queueing theory. In particu-lar, this assumption becomes a serious problem in the proposedsystem because the reentering customer causes a delay in trans-portation on the traveling lane. To solve this problem, in thispaper, approximation models that consider the e ff ect of walk-ing distance are proposed in Section 4.
3. Simulations
We set the τ in , σ in , τ s , and σ s to be 20, 1, 300 and 10, respec-tively, in this paper. The particle flux at the rightmost site wasmeasured in proportion to the number of parking sites and thenumber of total time steps to determine the condition of reach-ing the stationary state. Thereafter, we set the total number of3 𝑏 𝑘 BusyReservedOccupied
Figure 3: Dependence of the number of reserved / occupied sites and the numberof busy sites (the sum of reserved sites and occupied sites) on the di ff erentvalues of distribution parameter k between -10 and 10. parking sites N s to be 48 and set the number of total time stepsto be 2 , , / occupied sites and the number of busy sites (the sumof reserved sites and occupied sites) on the di ff erent values ofdistribution parameter k between -10 and 10. It was observedthat the number of busy sites N b increases with a gentle S-shapecurve as the parameter k decreases. Substantially, the increasein the number of busy sites N b is found to be determined onlyby the increase in reserved sites N r . On the other hand, thenumber of occupied sites N o remained constant during the sim-ulations. The result in Fig.3 indicates that the utilization of U servers in the classical M / M / c queueing theory corresponds notto the number of occupied sites N o , but to the number of busysites N b in the proposed system. This is because a parking sitebecomes accessible every time the previous busy time ends. Inthe next section, we investigate the relationship between N b andthe distribution parameter k of the exponential function.
4. Analysis
At the beginning of this study, we proposed a fundamentalmodel, which is simulated by the concept of D-Fork system.By considering the e ff ect of walking distance from the leftmostsite, the occupied time T i of the i − th parking site can be mod-eled as follows: T i = τ s + i ∆ l + α (15)Here, α is a constant parameter and ∆ l is the distance betweentwo parking sites.In the right-hand side of Eq.(15), the first term indicates thestaying time in a parking site. The second term indicates thetraveling time to the parking site. In this approximation level, we ignore the e ff ect of volume exclusion e ff ect in the secondterm; a particle is assumed to hop to the neighboring cell pera step. Thus, the velocity of the particle becomes 1 since thelength of a cell is set to 1. That is why the notation of velocitydoes not emerge in the second term. Instead, we introduce the α in the third term, assuming that the volume exclusion e ff ectcan be approximated as constant values in the target system.The maximum service rate µ max and the minimum servicerate µ min are obtained by substituting N s and 1 to Eq.(15), asfollows: µ max = T N s − (16) µ min = T − (17)The averaged value of the service rate of the system µ avr is ob-tained by calculating the arithmetic mean of Eq.(15), as follows: µ avr = (cid:40) τ s + N s N s (cid:88) i = ( i ∆ l + α ) (cid:41) − (18) = (cid:40) τ s + ∆ l N s + ∆ l + α (cid:41) − (19)Finally, the averaged number of busy sites N b is obtained bydividing Eq.(13) by Eq.(19), as follows: N b = λµ avr (20) = (cid:32) ∆ l τ in (cid:33) N s + τ in (cid:32) τ s + ∆ l + α (cid:33) (21)Equation (21) shows that the number of busy sites N b is a linearfunction of the number of sites N s . Figure 4 shows the simu-lation result of the dependence of the number of busy sites N b on the total number of sites N s , in the case of k =
0. It wasobserved that the break in line occurs at around N s =
17, whichis because all the sites are in use due to the lack of capacity ofsites when N s <
17. In comparison with Fig.3, the y-interceptvalue of the fitting line in Fig.4 corresponds to the value of thenumber of occupied sites N o . This is because the increment ofthe number of busy sites N b depends only on the increase inthe number of reserved sites N r . By fitting the line at N s > α is obtained to be 6.215as a fitting result. It was confirmed from the red colored line inFig.5 that the simulation results are bounded between the max-imum case (the dotted line) and the minimum case (the dashedline). Besides, the averaged case of classical M / M / c with theparameter α is found to semi-experimentally correspond to thecase of k = On the basis of an assumption that the site usage distribu-tions obey the exponential function ke − kx , we correct Eq.(19)by replacing the arithmetic mean by the weighted average us-ing the exponential function ke − kx . The number of busy sites4 𝑏 𝑁 𝑠 Figure 4: Dependence of the number of busy sites N b on the total number ofsites N s in the case of k = N b is calculated as follows: N b = τ s τ in + τ in (cid:80) N s i = ( i ∆ l + α ) E i (cid:80) N s i = E i (22) E i = k · exp( − k iN s ) (23)The red colored line and blue colored line in Fig.5 show thecomparison of simulation results and the estimated values ob-tained using Eq.(22), respectively. It was confirmed that thefeature of S-shape curve is observed in both simulations andapproximations. However, the number of busy sites N b calcu-lated by Eq.(22) becomes overestimated / underestimated at theboth sides of k < k > ff erent values of parameter k between −
10 and 10.Obviously, the shape of each distribution is di ff erent from thatof the exponential function. It should be noted that the reasonwhy the distribution gets slightly biased to the leftmost site inthe case of k = In this section, we describe the site usage distributionsby introducing the birth-death process for walking direction.Namely, the particles in transport before stopping at a site onthe parking lane (red colored particles in Fig.1) are regarded as“surviving”. On the contrary, an event in which a particle stopsat a site indicates the “death” of the particle.We define a random variable X , which donates a positionon the traveling lane, f ( x ) is defined as the probability density 𝑁 𝑏 𝑘 Figure 5: Comparison of the simulation results, the estimated values obtainedusing Eq.(22), the maximum values by dividing Eq.(13) by µ max , the mini-mum values by dividing Eq.(13) by µ min , the averaged values obtained by usingEq.(19), for di ff erent values of distribution parameter k between -10 and 10. function of X . From these definitions, the cumulative densityfunction F ( x ) of f ( x ) can be expressed as follows: F ( x ) : = P ( X ≤ x ) = (cid:90) x f ( x ) dx (24) F ( x ) represents the probability of “death” at position x . Con-versely, we define S ( x ) as the probability of surviving at posi-tion x , as follows: S ( x ) : = − F ( x ) = P ( X > x ) (25)Besides, a hazard function h ( x ) is defined as follows: h ( x ) : = lim ∆ → ∆ · P ( x < X < x + ∆ | X > x ) (26) h ( x ) is the probability density that a particle stops at a site at theposition between x and x +∆ . Equation.(26) can be transformedas follows: = lim ∆ → ∆ · P { ( x < X < x + ∆ ) ∩ ( X > x ) } P ( X > x ) (27) = lim ∆ → ∆ · P ( x < X < x + ∆ ) P ( X > x ) (28)From Eq.(24) and Eq.(25), = P ( X > x ) lim ∆ → F ( x + ∆ ) − F ( x ) ∆ (29) = F (cid:48) ( x ) S ( x ) (30) = − S (cid:48) ( x ) S ( x ) = − ddx (log( S ( x ))) (31)Here, we impose an initial condition S (0) = 𝑘 = 𝑘 = 𝑘 = 𝑘 = 𝑘 = 𝑘 = 𝑘 = -10 𝑘 = -1 𝑘 = -2 ****** 𝑘 = -3 𝑘 = -41 Index of parking sites
Figure 6: All the distributions for di ff erent values of parameter k between − always becomes 1. Then, the di ff erential equation Eq.(31) canbe soloved as follows: S ( x ) = exp( − H ( x )) (32)Here, we introduce the H ( x ), as follows: H ( x ) : = (cid:90) x h ( x ) dx (33)We obtain F ( x ) and f ( x ) from Eq.(32) as follows: F ( x ) = − exp( − H ( x )) (34) f ( x ) = h ( x ) · exp( − H ( x )) (35) f ( x ) corresponds to the probability distribution of site usagesince F ( x ) is the probability of death at position x . We havesuccessfully obtained a general formula for site usage distribu-tion in the proposed system. We introduce extreme statistics inSection 4.3.3 to determine the specific formula of H ( x ). The derivation of Eq.(34) lacks the information of orderstatistics of the random variable X . We introduce the conceptof order statistics to the proposed system in this section, as apreliminary work for the approximation by extreme statistics inSection 4.3.3.Let us consider the situation that a single particle is insertedfrom the queue to the leftmost site at certain intervals of arrivaltime during the total n time steps. We name the i − th insertedparticle to the leftmost site simply as “i-th particle”. We definethe random variables X i , which indicates the position that i − thparticle stops at during its travel. As similarly in the previoussection, the i − th particle is judged as “death” when X i ≤ x . Onthe contrary, the particle is judged as “surviving” when X i > x .An identical cumulative density function A ( x ) of X , X , . . . , X n is expressed, as follows: A ( x ) : = P ( X i ≤ x ) , ( i = , , . . . , n ) (36) o rightmost siteleftmost site space time 𝑥 𝑋 𝑋 𝑋 𝑋 𝑋 Independent variables 𝑋 , 𝑋 , 𝑋 , 𝑋 , 𝑋 o rightmost siteleftmost site space 𝑥 𝑋 𝑋 𝑋 𝑋 𝑋 Order statistics = 𝑋 , 𝑋 , 𝑋 , 𝑋 , 𝑋 reordering by positions 𝑋 (1:5) , 𝑋 (2:5) , 𝑋 (3:5) , 𝑋 (4:5) , 𝑋 (5:5) Figure 7: Relationship between the independent variables and the order statis-tics in case of n = The order statistics of X , X , . . . , X n , which is obtained byrearranging the X , X , . . . , X n in an ascending order, is repre-sented as follows: X (1: n ) ≤ X (2: n ) ≤ · · · ≤ X ( n : n ) (37)From the definition of the order statistics, it is obvious that the i − th order statistic X ( i : n ) corresponds to the i − th largest originalindependent variables. The relationship between the originalindependent variables and the order statistics in case of n = A X ( m : n ) ( x ) of the m − th statistic X ( m : n ) is defined as follows: A X ( m : n ) ( x ) : = P ( X ( m : n ) ≤ x ) (38)Considering the fact that X ( i : n ) corresponds to the i − th largestoriginal independent variables, it can be said that Eq.(38) indi-cates the probability that at least m variables of X , X , . . . , X n become equal or less than the position x ( = the state of “death”).By using the probability P j that exactly j variables of X , X , . . . , X n becomes equal or less than the position x , theright-hand side of Eq.(38) is represented as follows: P ( X ( m : n ) ≤ x ) = n (cid:88) j = m P j (39)The probability P i is further decomposed as follows; there are n C j di ff erent combinations of j variables from X , X , . . . , X n .In each case, j variables become “death” with the probability A ( x ) and that of n − j variables become “survival” with theprobability 1 − A ( x ); the cumulative density function A X ( m : n ) ( x )is represented by using Eq.(38) and Eq.(39), as follows: A X ( m : n ) ( x ) = n (cid:88) j = m n C j A ( x ) j (1 − A ( x )) n − j (40)Now we obtain the precise expression of Eq.(38). Equa-tion (40) includes all the possible patterns of particle arrivalsfor the time direction during the total n time steps.6 (𝑍 𝑛 ≤ 𝑥) 𝑃(𝑌 𝑛 ≤ 𝑥) (a) (b) Figure 8: Schematic view of the distributions of two di ff erent extreme orderstatistics. Unfortunately, it is di ffi cult to derive the probability distri-bution of site usage directly from Eq.(40) because the cumu-lative density function A ( x ) at each time step is unknown. Tosolve this problem, we propose to approximate the site usagedistribution by the asymptotic distribution of the distribution ofextreme order statistics in the next section. The maximum order statistics Z n and the minimum orderstatistics Y n are defined, respectively, as follows: Z n : = X ( n : n ) = max { X , X , . . . , X n } = max ≤ i ≤ n X i (41) Y n : = X (1: n ) = min { X , X , . . . , X n } = min ≤ i ≤ n X i (42)In this paper, we propose to approximate the cumulativeprobability distribution of site usage in Eq.(24) by the distribu-tion of extreme order statistics. Here, we have two candidatesof P ( Z n ≤ x ) and P ( Y n ≤ x ).We are able to say that the selection of P ( Y n ≤ x ) is appro-priate for the approximation of Eq.(24) considering the phys-ical meaning of these extreme order statistics. P ( Z n ≤ x ) de-scribes the probability that not a single X i becomes larger thanthe position x during the time step n , as shown in Fig.8(a). Be-cause the situation depicted in Fig.8(a) seldom occurs in theproposed system, replacing the random variables X in Eq.(24)by the maximum order statistics of Z n is not appropriate. On thecontrary, as shown in Fig.8(b), P ( Y n ≤ x ) describes the proba-bility that at least one of X i becomes smaller than the position x during time step n ; therefore, the asymptotic distribution ofminimum order statistics of Y n is suitable for describing the be-haviors of the proposed system, compared to the former case.We approximate the cumulative density distribution of siteusage in Eq.(24) by the distribution of minimum order statistics Y n , as follows: F ( x ) ≈ P ( Y n ≤ x ) (43)For adequate large n , it is known that the distribution of min-imum order statistics Y n asymptotic to the following extremevalue distribution M ( x ) in case that the random variables of X , X , . . . , X n follow exponential distributions[26, 27]: P ( Y n ≤ x ) → M (cid:32) x − ˜ c ˜ b (cid:33) (44) = − exp (cid:34) − exp (cid:32) kx − cb (cid:33)(cid:35) (45)Here, (˜ c , ˜ b ) are normalizing constants, which are selected toconvert the location and scale so that the extreme value distri-bution M does not diverge and degenerate. The ( c , b ) is ( k ˜ c , k ˜ b ),respectively. For detail description on the derivation of Eq.(45),see the Appendix.Now we obtain the distribution F ( x ) of the proposed system,as follows: F ( x ) = − exp (cid:34) − exp (cid:32) kx − cb (cid:33)(cid:35) (46)The probability density function f ( x ) is obtained as follows: f ( x ) = kb · exp (cid:32) kx − cb (cid:33) · exp (cid:34) − exp (cid:32) kx − cb (cid:33)(cid:35) (47)From Eq.(47), the cumulative hazard function H ( x ) is found tobecome an exponential function: H ( x ) = exp (cid:32) kx − cb (cid:33) (48)The selection of Y n is validated from the point of mathemat-ical derivation. If we select Z n , the right-hand side of Eq.(46)becomes exp[ − e − ( kx − f ) / g ]. This description contradicts with theformula obtained in Eq.(34). In this case, the relationship be-tween Eq.(34) and Eq.(35) is not satisfied.It is not easy to mathematically derive the constant parame-ters of ( c , b ) of minimum order statistics Y n , we determine theseparameters by fitting the simulation results in the next section. / M / c queueing model Let us get back to the subject of queueing theory. We attemptto correct the weighted calculation in Eq.(22) by replacing theexponential function by the fitting function of the simulationresults. We adopt Eq.(47) as the fitting function, admitting thetransformation of the scale of Eq.(47) by using constant param-eter a , as follows: f FIT ( x ) = a kb · exp (cid:32) kx − cb (cid:33) · exp (cid:34) − exp (cid:32) kx − cb (cid:33)(cid:35) (49)Figure 9 shows all the cases of exponential distributions fittedby Eq.(49) for di ff erent values of parameter k between −
10 and10. The dashed red colored lines indicate fitting results by theleast squares method. Figure 10 shows the dependence of thechi-square of fitting results in Fig.9 on the di ff erent values ofparameter k . Obviously, in Fig.10, it was observerd that theaccuracy of curve fitting deteriorates as the bias to the right / leftside increases. The reason for this is interpreted as follows.As the bias to the right / left side increases, congestion occurs7 igure 9: All the distributions fitted by Eq.(49) for di ff erent values of parameter k between −
10 and 10. 𝜒 𝑘 Figure 10: Dependence of the chi-square of fitting results in Fig.9 on the di ff er-ent values of parameter k . in the neighboring area of the rightmost / leftmost site. Becausethe e ff ect of congestion is not considered in the deviation ofEq.(49), the di ff erence at both sides of the edges emerges.We correct the weighted calculations of the proposed queue-ing model in Eq.(22) by replacing the weighted function withthe function in Eq.(49), as follows: N b = τ s τ in + τ in (cid:80) N s i = ( i ∆ l + α ) E i (cid:80) N s i = E i (50) E i : = f FIT ( iN s ) (51)Figure 11 shows a comparison of (a) the simulation resultsin Fig.5 with (b) the estimated values obtaind using Eq.(22)and (c) the estimated values obtained using Eq.(50). It wasconfirmed that our proposed model shows a good agreementwith the simulation results compared to the model exhibited in 𝑁 𝑏 𝑘 (a)(c)(b) Figure 11: (a) the simulation results in Fig.5, (b) second-order proposed modelexhibited in Eq.(22), and (c) third-order proposed model exhibited in Eq.(50).
Eq.(22). This result indicates that the proposed method, whichestimates the service rate µ s by using weighted calculations ofthe site usage distributions, is an e ff ective approach under cer-tain conditions.
5. Conclusion
We introduced a totally asymmetric simple exclusion processon a traveling lane equipped with a queueing system and func-tions of site assignments along the parking lane. In this study,we investigated the relationship between the utilization of park-ing sites and the e ff ect of site assignments in the proposed sys-tem. The contributions of this study are as follows:We proposed an approximation model to describe the site us-age distributions of the proposed system on the basis of birth-death process for the spatial direction and extreme statisticsfor the time direction. The specific formula in the case wherethe random variables follow exponential distributions are de-scribed. In addition, our proposed M / M / c queueing model,whose service rate is determined by the weighted calculation ofsite usage distributions, shows good agreement with the simual-tion results.As mentioned in the introduction, the major scope of the cur-rent research is to describe the relationship between the utiliza-tion of parking sites and the e ff ect of site assignments in theproposed system. Accordingly, we obtained insightful resultsfrom the findings of this study. Acknowledgements
A. Asymptotic distributions of the distributions of extremeorder statistics
The distributions of maximum order statistics Z n and mini-mum order statistics Y n are represented as follows: P ( Z n ≤ x ) = P ( X ( n : n ) ≤ x ) = A X ( n : n ) ( x ) (52) P ( Y n ≤ x ) = P ( X (1: n ) ≤ x ) = A X (1: n ) ( x ) (53)For adequate large n , the distributions of these two extremeorder statistics are assumed to asymptotic to the extreme valuedistributions, respectively, as follows: P ( Z n ≤ x ) → G (cid:32) x − a n b n (cid:33) (54) P ( Y n ≤ x ) → M (cid:32) x − c n d n (cid:33) (55)Here, ( a n , b n ) are normalizing constants, which are selected toconvert the location and scale of G so that the extreme value dis-tribution G does not diverge and degenerate. The same is true of( c n , d n ). The assumption of the existance of G and Eq.(54) arevalidated on condition that Eq.(57) is satisfied [26, 27]. If theyare validated, the asymptotic distribution M ( x ) of minimum or-der statistics Y n is obtained from the following relationship: M ( x ) = − G ( − x ) (56) B. Trinity Theorem
A population distribution F is assumed to belong to a domainof attraction of an extreme value distribution G ; this assumptionis denoted as F ∈ D ( G ). R.A. Fisher and L.H.C Tippett[28]mathematically proved the following relationship for maximumorder statistics Z n : F ∈ D ( G ) ⇔ lim n →∞ F n ( a n x + b n ) = G ( x ) , a n > , b n ∈ R (57)After considerable e ff orts, mathematicians Fr´echet[29], R. A.Fisher and L.H.C Tippett[28], and Gnedenko[30] proved a no-table fact that only three types of extreme distributions exist,which are as follows:Gumbel : G ( x ) = exp[ − exp( − x )] , x ∈ R (58)Fr´echet : G ( x ) = exp( − x − α ) , x ≥ , α > G ( x ) = exp[ − ( − x ) α ] , x ≤ , α ≥ F isasymptotic to one of the three kinds of extreme distributionslisted from Eq.(58) to Eq.(60), on the condition that the relation F ∈ D ( G ) is satisfied. C. The extreme value distributions of an exponential distri-bution
The asymptotic distribution for the case in which the randomvariables of X , X , . . . , X n follow exponential distributions isobtained, as follows. A cumulative exponential function is writ-ten as follows: F ( x ) : = − exp( − kx ) (61)Here, we use the following indentity equation: F n ( a n x + b n ) = (cid:40) + − n [1 − F ( a n x + b n )] n (cid:41) n (62)By selecting a n = b n = k − log( n ), − n [1 − F ( a n x + b n )] = − exp( − kx ) , x ≥ . (63)By substituting Eq.(63) into Eq.(62),lim n →∞ F n ( a n x + b n ) = lim n →∞ (cid:32) + − e − kx n (cid:33) n (64) = exp[ − exp( − kx )] (65)From Eq.(57), we obtain the expression of G ( x ), as follows: G ( x ) = exp[ − exp( − kx )] (66)Equation (65) indicates that the asymptotic distribution G ( x )of maximum order statistics Z n , when the random variables X , X , . . . , X n follow an exponential distribution, belongs to thefamily of Eq.(58) in the Trinity Theorem.From the relationship in Eq.(56), we obtain the expression of M ( x ), as follows: M ( x ) = − exp[ − exp( kx )] (67)By substituting Eq.(67) into Eq.(55), we obtain as follows: P ( Y n ≤ x ) → M (cid:32) x − c n d n (cid:33) (68) = − exp (cid:34) − exp (cid:32) kx − ˜ c n ˜ d n (cid:33)(cid:35) (69)Here, ˜ c n and ˜ d n are kc n and kd n , respectively. References [1] A. K. ERLANG, The theory of probabilities and telephone conversations,Nyt. Tidsskr. Mat. Ser. B , 33 (1909).[2] D. Helbing, Tra ffi c and related self-driven many-particle systems, Rev.Mod. Phys. , 1067 (2001).[3] J. Walraevens, T. Demoor, T. Maertens, and H. Bruneel, Stochasticqueueing-theory approach to human dynamics, Phys. Rev. E , 021139(2012).[4] P. Blanchard and M.-O. Hongler, Modeling human activity in the spiritof barabasi’s queueing systems, Phys. Rev. E , 026102 (2007).[5] S. Choubey, Nascent rna kinetics: Transient and steady state behavior ofmodels of transcription, Phys. Rev. E , 022402 (2018).[6] C. Baker, T. Jia, and R. V. Kulkarni, Stochastic modeling of regulation ofgene expression by multiple small rnas, Phys. Rev. E , 061915 (2012).[7] C. Arita, Queueing process with excluded-volume e ff ect, Phys. Rev. E , 051119 (2009).
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