Effect of self-deflection on a totally asymmetric simple exclusion process with functions of site-assignments
EEffect of self-deflection on a totally asymmetric simpleexclusion process 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 study proposes a model of a totally asymmetric simple exclusion processon a single channel lane with functions of site-assignments along the pitlane.The system model attempts to insert a new particle to the leftmost site ata certain probability by randomly selecting one of the empty sites in thepitlane, and reserving it for the particle. Thereafter, the particle is directedto stop at the site only once during its travel. Recently, the system was de-termined to show a self-deflection effect, in which the site usage distributionbiases spontaneously toward the leftmost site, and the throughput becomesmaximum when the site usage distribution is slightly biased to the right-most site, instead of being an exact uniform distribution. Our exact analysisdescribes this deflection effect and show a good agreement with simulations.
Key words:
Totally Asymmetric Simple Exclusion Process, DeflectionEffect, Statistical Mechanics, Assignment Problems, Multi-Particle Physics
Email addresses: [email protected] (Satori Tsuzuki), [email protected] (Daichi Yanagisawa), [email protected] (Katsuhiro Nishinari)
Preprint submitted to Elsevier September 25, 2018 a r X i v : . [ n li n . C G ] N ov . Introduction The totally asymmetric simple exclusion process (TASEP) has been ex-tensively studied in the field of non-equilibrium statistical mechanics. Byemploying the simple characteristic that a particle hops in a single directionwith the constraint of volume exclusion effect, the TASEP has been appliedto many practical problems (e.g. molecular motor transport[1, 2], granularflows [3, 4], and traffic system [5, 6, 7, 8, 9]). Recently, the applicabilityof the TASEP has been expanded by combining with the stochastic processmodels based on the M/M/1 queueing theories [10, 11].This study proposes a TASEP model for the problem of flows in a singlechannel lane with an adjacent pitlane under the open boundary condition.The subject of this study is that the proposed model has function of site-assignments, according to which the system tries to insert a new particle tothe leftmost site at a certain probability; if successful, the system randomlyselects one of the sites in the pitlane and reserves it for the particle to stopat once during its travel. This proposed model is not only scientificallyinteresting but also applicable to problems in engineering because such atype of mechanism can be observed in many practical cases. For example,during the taxiing of an aircraft in airport ground transportations, the airporttraffic controller selects one of the empty spots on the apron and assigns it tothe aircraft in the rapid taxiway coming from the runway, and then instructsthe aircraft to stop at the selected spot. The mechanism of aircraft taxiingis exactly the same as the applied case of the model proposed in this study.Other real-world examples could include the vehicular traffic in a parking lotand the problem of air boarding [12, 13, 14, 15].2everal studies have reported on the TASEP by considering the absorp-tion lane (e.g. multiple lanes[16, 17, 18] and TASEP with Langmuir Kinetics[17, 18, 19, 20, 21, 22, 23]). The proposed model can be said to be classifiedinto the same category in a wider sense, expect the function of site assignmentin this study is the distinguishing factor compared to the models reportedin the previous studies. The major scope of the current research is to clarifythe relationship between the site-assignments and system properties of theproposed model; this has not been focused on in the aforementioned studies.The reminder of this paper is structed as follows. Section 2 describesthe details of the proposed model. Section 3 presents the investigation ofdependence of system properties on each parameter through simulations. InSection 4, the deflection effect of the proposed model is described through ourexact analysis. Section 5 summarizes our results and concludes this paper.
2. Model
A schematic view of our proposal TASEP model is depicted in Fig.1 . Thesystem consists of two parallel lanes: a single channel lane and an pitlane.Each lane is composed of L sites. We adopted the parallel update methodin which all the particles are simultaneously updated based on the followingrules: • Particle Insertion
The system tries to insert a new particle to the leftmost site with theprobability α . However, if the other particle exists in the leftmost site,the attempt is blocked. In contrast, if the attempt is successful, one ofthe empty sites of the pitlane is reserved for the particle. Thereafter,3 igure 1: Schematic view of the proposed model. the particle is directed to stop at the site of the pitlane only once duringits travel. • Site Management
The state of a site in the pitlane can be expressed in three ways: “oc-cupied,” “reserved,” and “unreserved.” The “occupied” state indicatesthat the site has already stored a particle. The “reserved” state indi-cates that the site has already been designated for one of the particlesinserted to the leftmost site. The “unreserved” state indicates that thesite is not occupied or reserved by any particle. In site-assignments,the system sets serial numbers to unreserved sites. In the illustrationshown in Fig.1, indices in the range of 0 to N unr − • Site Assignment
This study describes three methods of site-assignment: (a) uniformrandom distribution; (b) random distribution, which is biased by aleftmost exponential function of λe − λx ( x ≥ , λ > λ increases by assigning serial numbers to theunreserved sites in the reverse order. In this paper, we term the (b)and (c) methods simply as “bias toward the leftmost site” and “biastoward the rightmost site.” Note that the inverse transform sampling(ITS)[24, 25] was adopted to generate biased random numbers used in(b) and (c). • Transportation of Particles
The particle in the channel lane hops to the right adjacent site witha probability of 1 in case the target site is empty. Here, the stateof particles can be expressed in three ways: “mobilization,” “stand-by,” and “re-mobilization.” The “mobilization” state indicates thatthe particle is in transport before stopping at the designated site of thepitlane. The “stand-by” state indicates that the particle is currentlystopping at the site of the pitlane. Futher, the “re-mobilization” stateindicates that the particle is in transport after exiting the pitlane. InFig.1, the states are expressed as the red, yellow, and blue coloredcircles, respectively. 5
Pit-in of Particles
The particle in the channel lane hops to the lower adjacent site in thepitlane with the probability of 1 in case the index of the site, which isreserved to the particle, corresponds to that of the current site. • Release of Particles
The particle in the pitlane hops to the upper adjacent site in the channellane with probability γ in case the target site in the channel lane isempty. The particle in the pitlane has the right of priority accessagainst the particle in the channel lane to hop to the same site. • Elimination of Particles
The particle in the channel lane is eliminated from the system at thenext step after hopping to the rightmost site.
3. Simulations
By using the total number of time steps N step , and total number of par-ticles that exit from the rightmost site during simulation, N out , throughput Q from the rightmost site is defined as follows: Q = N out N step (1) To determine the condition of reaching stationary state, the dependenceof throughput Q on each parameter is primarly investigated using uniformrandom distribution. The left and right sides in Fig.2 show the dependenceof throughput Q on the number of sites and number of time steps with the6 igure 2: Dependence of throughput Q on the number of sites (left side) and number oftime steps (right side) in case of setting the probability α to the different twenty values inthe range of 0 .
05 to 1 . L in the right side to 100. change in probability α from 0 .
05 to 1 .
0, respectively. Note that the numberof sites L on the right side in Fig.2 is fixed to 100.Regarding the dependence on the number of sites, it is observed that thescalability of Q slightly improves even as the number of sites increases tomore than 800. Regarding the dependence on time step, the scalability of Q gets saturated at least from 10 , Q on probabilities α and γ . It isconfirmed that throughput Q turns sluggish and reaches a plateau area asprobability α increases, whereas only a slight increase is observed with theincrease in probability γ .Figure 4 shows the dependence of throughput Q on probability α in caseof being (b) bias toward the leftmost site (in the left column) and (c) biastoward the rightmost site (in the right column) respectively. Note that the7 igure 3: Dependence of throughput Q on probabilities α and γ . result in case of using (a) uniform random distribution is included on bothsides in Fig.4 for comparison. The difference among simulation results wasobserved to be more obvious in the plateau area than the growth area (Thedependence of throughput Q on the distribution parameter λ is explained inthe next section).In response to these results, we setup the target problem as follows. Thenumber of sites L is set to be 1 , α and γ were set to be 1 . .
5, respectively. Q on parameter λ Figure 5 shows the dependence of throughput Q on distribution parame-ter λ of an exponential function. In the case of (b) bias toward the leftmostsite and (c) bias toward the rightmost site the results are displayed at the8 igure 4: Dependence of throughput Q on probability α in case of being (b) biased towardthe leftmost site (left column) and (c) biased toward the rightmost site (right column). right and left of zero, respectively. For the sake of easy view of (c) in a singlefigure, parameter λ is expressed by multiplying a negative sign in the leftpart, as shown in Fig.5. Moreover, the result of (a) uniform random distri-bution is shown at zero for comparison. Throughput Q gradually increasesas parameter λ decreases, and reaches its maximum when λ = −
1, whichcorresponds to the case of (c), in which parameter λ = 1, and thereafterthroughput Q decreases. Here, two important observations are made. First,throughput Q improves by assigning the sites closer to the rightmost site toparticles by using (c) with λ = 1, rather than using (a) uniform random dis-tribution; Second, the gradient of the line in (c) becomes moderate comparedto that in (b).To understand the reasons for these phenomena, it is reasonable to inves-tigate the site usage distribution because the amount of usage at each siteis equal to the occurrence count of the transport delay in this system. This9 igure 5: Dependence of throughput Q on distribution parameter λ . is because the delay of particles for the traveling direction is only causedby encounters between two particles in different lanes. The particle in thechannel lane must stop at the current site when the particle in the pitlaneexits the site and hops to the adjacent right site.Figure 6 shows the site usage distribution according to each simulationresult in Fig.5. The results represented by each colored line correspond tothose in Fig.4 represented by the same color. Figure 6 presents an importantphenomenon, which is the main topic of this paper: the spontaneous distri-bution of the site usage biases toward the leftmost site even when using (a)uniform random distribution, as shown in the center part in Fig.6.In case of (b) bias toward the leftmost site, the site usage distribution isdirectly reflected by the scale of the exponential function of λe − λx accord-ing to each parameter λ . The traffic congestion amplified by parameter λ igure 6: Site usage distribution according to each simulation result in Fig.5. provides a good explanation for the decline of throughput Q , as shown inthe right part in Fig.5. In contrast, in case of (c) bias toward the rightmostsite, the distribution shape differs from the symmetric case of (b). This isbecause each distribution is biased toward the leftmost site owing to thespontaneous bias. Consequently, the traffic congestion around the rightmostsite is partly reduced in case of λ = 1, and therefore throughput Q increases.Thereafter, throughput Q decreases because of the traffic congestion aroundthe rightmost site amplified by parameter λ > Q and the spontaneous bias.
4. Analysis
This section presents the derivation of an approximation based on theMarkov chain model. As it is difficult to consider all the L sites, we approx-imated the proposed system by using four sites composed of two upper sitesin the channel lane and two lower sites of the pitlane located at the right edge11f two lanes. As two kinds of particles, that is, the mobilization (red coloredparticles), and re-mobilization (blue colored particles) particles exist in thetwo sites in the channel lane, as mentioned in Section 2, each of the two sitesin the channel lane possibly has three states: “unoccupied,” “occupied bya red particle,” and “occupied by a blue particle.” Moreover, each of thetwo sites in the pitlane has possibly two states “occupied” or “unoccupied”because only a single kind of particle exists in the two sites in the pitlane.Therefore, state space S of this system is expressed as follows: S = { s ( i, j, k, l ) | i, j = 0 , , , k, l = 0 , } (2)Here, s is a state designated by group ( i, j, k, l ). Parameter i is the serialindex, which is set to the three states of the left site, j is the serial index setto the three states of the right site in the channel lane, k is the serial indexset to the two states of the left site, and l is the serial index set to two statesof the right site in the pitlane. Therefore, this system has 36 possible states,provided that the rules explained in Section 2 are not considered.Second, we define a transition probability P n ( i, j, k, l ), which is the proba-bility that a system state at time step n becomes s ( i, j, k, l ). Simultaneously,we define s nm and P nm as abbreviations of s ( i, j, k, l ) and P n ( i, j, k, l ), respec-tively. Here, index m is a serial number assigned to all the possible statesand is uniquely indicated by group ( i, j, k, l ).As our point of focus was in the stationary state, we set probability α to 1 .
0. Thus, by considering the rules in Section 2, the possible states arereduced from 36 to 20. Note that state space ˆ S spanned by these 20 statesis subspace of S . Figure 7 shows the images of all the 20 states in the leftcolumn, and the resultant master equations obtained by investigating all the12 igure 7: The 20 possible states (left column) and master equations of the proposed system(right column). state transitions among these 20 states in the right column. Here, C R and C B are the probabilities that the red and blue colored particles, respectively,enter the left site of the two sites in the channel lane, and C L is the probabilitythat no particle enters the system. It is also assumed in the master equationsthat the probabilities of C R , C B , and C L become constant at the stationarystate and these constant probabilities must meet the following normalization: C R + C B + C L = 1 (3)13urthermore, transition probability P nm ( m = 1 , , , . . .
20) must meet thefollowing normalization condition: (cid:88) m =1 P nm = 1 (4)In transitions from s n , s n , and s n , the red colored particle is not allowedto flow into the system as both sites in the pitlane have already been occu-pied; thus, we assume another pair of constant probabilities that meet thecondition in which the probability of the red colored particle flowing into thesystem becomes zero: C B + C L = 1 (5)In the end, we introduce a constant probability q in the transition from s n to s n +13 and from s n to s n +15 in the master equations, as shown in Fig.7because only these two transitions have the selectivity of sites. It was as-sumed that a red particle hops to the lower-left and lower-right sites in thepitlane with probabilities q and 1 − q , respectively, in these transitions.The conversion of transition probability matrix ˇ P ( n ) spanned by state P nm ( m = 1 , , . . .
20) at time step n to matrix ˇ P ( n + 1) at time step n + 1 issimply written as follows: ˇ P ( n + 1) = M · ˇ P ( n ) (6)Here, M is a 20 ×
20 matrix constructed using the constant coefficients ofeach probability in the right column in Fig.7. At the stationary state, sincethe transition probability becomes constant, the system shows the followingrelationship: M · ˇ P ( n ) = ˇ P ( n ) (7)14quation (7) indicates that matrix M has an eigenvalue of 1 in case thesystem is in the stationary state. In this case, the normalized eigenvectorsof M correspond to the elements of ˇ P ( n ). L = 2Let us consider the case where the length of sites L is set to 2. In thiscase, since the system continuously tries to insert a new red particle into theleftmost site in the channel lane with probability 1, the constant coefficientof C R is always 1, and both C B and C L are always 0. In addition, C B mustbe set to 0 as it is not allowed for the blue colored particle to flow into thesystem in transitions from s n , s n , and s n in case the length L of sites is 2.Under these limitations, the eigenvector of matrix M is obtained as follows: P n P n P n P n P n P n P n P n P n P n P n P n P n P n P n P n P n P n P n P n = 35 q/ (334 − q ) × − q ) / q (1 − q ) /q (6 q + 16) / q /q (9 − q ) / q (32 − q ) / q /q (44 − q ) / q (6 q + 16) / q /q (18 − q ) / (35 q )00(19 − q ) / q (8)15e estimated the expected value E Q of throughput Q by summing up thestate in which a blue particle exists in the upper-right site in the channel lane,with a weighted coefficient obtained by each transition probability. Consid-ering s n , s n , s n , s n , s n , s n , and s n , the expected value E Q of throughput Q is obtained as follows: E Q = 95 − q − q (9)The dependence of E Q of throughput Q on distribution probability q isshown as the blue colored line in Fig.8. It is found that the E Q graduallyimproves as q decreases, implying that the throughput increases by preferen-tially assigning sites in the neighborhood of the rightmost site to the particlein the leftmost site.The red colored line with the square symbol in Fig.8 shows the depen-dence of throughput Q on probability q simulated using TASEP by changingprobability q from 0 to 1. The simulations were confirmed to show a goodagreement with our exact analysis. The fact that throughput Q improvesby assigning the sites in the pitlane closer to the right edge to particles isvalidated by both the exact analysis and simulations.Similarly, in the estimation of E Q , we estimated the expected value ofthe usage of the left site in the pitlane by summing up the state in thatyellow-colored particle existing in this left site, weighted by each transitionprobability. Considering s n , s n , s n , s n , s n , s n , s n , s n , and s n , the expectedvalue E LS of the site usage of the left site in the pitlane is estimated as follows: E LS = 104 + 4 q − q (10)Similarly, we estimated the expected value of the usage of the right site by16 igure 8: Dependence of the expected throughput E Q on distribution probability q . summing up the state in which a yellow-colored particle exists in the right sitein the pitlane, weighted by each transition probability. Considering s n , s n , s n , s n , s n , s n , and s n , the expected value E RS of the usage of the right site inthe pitlane is estimated as follows: E RS = 111 − q − q (11)Figure 9 shows the dependence of the expected site usage E LS of the leftsite in the pitlane (red colored line with the circle symbol), and the expectedsite usage E RS of the right site in the pitlane (blue colored line with thesquare symbol) on probability q . The expected site usage E LS of the leftsite was confirmed to increase more than the expected site usage E RS of theright site if probability q was set to 0 .
5, indicating that the amount of site17 igure 9: Comparison of the expected value E S of the site usage for left site (red coloredline) and right site (blue colored line) in the pitlane. usage is spontaneously biased toward the leftmost site. We termed this asthe “self-deflection effect.” In addition, the expected site usage E LS of the leftsite was confirmed to decline as probability q decreased, and become equalto the expected site usage E RS of the right site at approximately q = 0 . E RS of the right site increased more than E LS of the left site.The comparison with Fig.8 showed that throughput still improves evenafter the expected site usage E RS of the right site increases more than the ex-pected site usage E LS of the left site. This shows that the maximum through-put is achieved when considering the slight bias toward the rightmost site,rather than the uniform distribution.18n more detail, this deflection effect is recognized as follows: no matterhow we add external forces to change the site usage distribution biased towardthe rightmost site, such external force only affects the transitions from s n to s n +13 and from s n to s n +15 in the master equations as probability q onlyemerges in these transitions. Owning to the other states, the site usagedistribution spontaneously biases toward the leftmost site. In other words,this highlights the trade-off relationship between the “self-deflection effect”and the bias given by external forces. In both exact analysis and simulations,the throughput was found to improve by the addition of external bias towardthe opposite direction of the self-deflection.The self-deflection effect can be qualitatively explained by consideringtraveling times of particles as follows. If we set L to 2, the traveling time tothe right site in the pitlane increases more than that to the left site in thepitlane; two steps are necessary for particles to travel toward the right site inpitlane, whereas only one step is needed for them to travel toward the left sitein pitlane. Since it is not allowed for different two particles to book the samesite, the disabled time of reservations for the left site in the pitlane becomesshorter than that for the right site in the pitlane. Thus, the turnover of siteusage of the left site becomes larger than that of the right site. Subsequently,the amount of the site usage for the left site becomes larger than that for theright site; the system spontaneously deflects. The relationship between thesystem characteristic value of throughput and site usage has not been focusedon in previous studies; this is quantitively described by exact analysis andsimulations in this paper.The system discussed in this paper is scientifically interesting from the19iewpoint of multi-particle dynamics. For example, it might be possible tofind similar phenomena in the position-sensitive detection system of heavyparticle beams for tumor therapies[26]. The particles reduce their kineticenergies in human tissues and emit pairs of annihilation gamma rays whenthey stop. By detecting these rays through positron emission tomography,it becomes possible to estimate the radioactive dose-distribution[27, 28]. Asthe difference of traveling times of particles affects the detection efficiencyat each position in many cases, the self-deflection effect might be observedin the dose-distribution. However, several possibilities exist observing thisdeflection effect in any other multi-particle system.
5. Conclusion
We introduced a TASEP model on a single channel lane with functionsof site-assignments along the pitlane. In this paper, we investigated therelationship between the site-assignments and the characteristic values ofthe proposed system. The contributions of this study are as follows:The proposed system shows a “self-deflection effect.” The distributionof site usage biases spontaneously toward the leftmost site, and thereforethroughput improves by the addition of the bias toward the rightmost site.The throughput becomes maximum when the site usage distribution becomesslightly biased toward the rightmost site, rather than the exact uniform dis-tribution. These findings are validated in both our exact analysis and simu-lations. This effect has been newly found in this study.As mentioned in the introduction, the major scope of this research is toclarify the relationship between the site-assignments and system properties20f the proposed model under open boundary conditions. Accordingly, weobtained beneficial insights from the findings of this study.
Acknowledgements
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