2.2t t 3 =1.2t+0.035' t= D, = f ? t 3 dt f ' 2 dt

Transcription

1 5.5 Queuing Theory and Traffic Flow Analysis 159 EXAMPLE 5.8 After observing arrivals and departures at a highway toll booth over a 60-minute tim e period, an observer notes that the arrival and departure rates (or service rates) ar e deterministic, but instead of being uniform, they change over time according to a know n function. The arrival rate is given by the function A(t) = ?, and th e departure rate is given by du(t) = t, t is in minutes after the beginning of the observation period and A(t) and,u(t) are in vehicles per minute. Determine the total vehicle delay at the toll booth and the longest queue, assuming D/D/1 queuing. Note that this problem is an example of a time-dependent deterministic queue because th e deterministic arrival and departure rates change over time. Begin by computing the time to queue dissipation by equating vehicle arrivals and departures : ? dt = f l tdt 2.2t t 3 =1.2t+0.035' t= 0 which gives t = 61.8 minutes. Therefore, the total vehicle delay (the area between the arriva l and departure functions) i s D, = f ? t 3 dt f ' 2 dt =1.1? ' ' 3 Jr = (61.8) (61.8) (61.8) 2 = veh-min The queue length (in vehicles) at any time t is given by the functio n Q(t) = f e 2 dt t dt = t t Solving for the time at which the maximum queue length occurs, 'Q(t)= _ ' = 0 dt t = mi n Substituting with t = minutes gives the maximum queue length :

2 160 Chapter 5 Fundamentals of Traffic Flow and Queuing Theory Q(39.12) = t t 2 +t = (39.12) (39.12) = ve h MID/1 Queuin g The assumption of exponentially distributed times between the arrivals of successiv e vehicles (Poisson arrivals) will, in some cases, give a more realistic representation of traffic flow than the assumption of uniformly distributed arrival times. Therefore, th e M/D/1 queue (exponentially distributed arrivals, deterministic departures, and on e departure channel) has some important applications within the traffic analysis field. Although a graphical solution to an MID/1 queue is difficult, a mathematical solutio n is straightforward. Defining a new term (traffic intensity) for the ratio of averag e arrival to departure rates as p = traffic intensity, unitless, 2 = average arrival rate in vehicles per unit time, an d,u = average departure rate in vehicles per unit time, 2 p = (5.27) and assuming that p is less than 1, it can be shown that for an M/D/1 queue th e following queuing performance equations apply : p p 2 = (5.28) 2(1 p ) w= P 2,1 ( 1 p ) (5.29 ) t 2 p (5.30) 2/1(1 p) Q = average length of queue in vehicles, w = average waiting time in the queue, in unit time per vehicle, t = average time spent in the system (the summation of average waiting time in th e queue and average departure time), in unit time per vehicle, an d Other terms are as defined previously.

3 5.5 Queuing Theory and Traffic Flow Analysis 161 It is important to note that under the assumption that the traffic intensity is les s than 1 (2 <,u), the D/D/1 queue will predict no queue formation. However a queuing model that is derived based on random arrivals or departures, such as the M/D/ 1 queuing model, will predict queue formations under such conditions. Also, note tha t the M/D/ l queuing model presented here is based on steady-state conditions (constant average arrival and departure rates), with randomness arising from the assume d probability distribution of arrivals. This contrasts with the time-varying deterministi c queuing case, presented in Example 5.8, in which arrival and departure rates changed over time but randomness was not present. EXAMPLE 5.9 Consider the entrance to the recreational park described in Example 5.7. However, let the average arrival rate be 180 veh/h and Poisson distributed (exponential times betwee n arrivals) over the entire period from park opening time (8 :00 A.M.) until closing at dusk. Compute the average length of queue (in vehicles), average waiting time in the queue, an d average time spent in the system, assuming M/D/1 queuing. Putting arrival and departure rates into common units of vehicles per minute give s and 180 veh/h = 3 veh/min for all t 60 min/ h _ 60 s/min = 4 veh/min for all t 15 s/veh A 3 veh/mi n p= = =0.7 5 ft 4 veh/mi n For the average length of queue (in vehicles), Eq is applied : Q _ 2( ) =1.125veh For average waiting time in the queue, Eq give s w= 2(4)( ) = min/veh For average time spent in the system [queue time plus departure (service) time], Eq i s used :

4 162 Chapter 5 Fundamentals of Traffic Flow and Queuing Theory t _ 2(4)( ) = min/veh or, alternatively, because the departure (service) time is 1/,u (the 0.25 minutes it takes th e park attendant to distribute the brochure), = = min/ve h M/M/1 Queuing A queuing model that assumes one departure channel and exponentially distribute d departure times in addition to exponentially distributed arrival times (an M/M/1 queue) is applicable in some traffic applications. For example, exponentiall y distributed departure patterns might be a reasonable assumption at a toll booth, wher e some arriving drivers have the correct toll and can be processed quickly, and other s do not have the correct toll, producing a distribution of departures about some mea n departure rate. Under standard M/M/1 assumptions, it can be shown that th e following queuing performance equations apply (again assuming that p is less than 1) : (5.31 ) 2 w=,u (,u 2 ) (5.32 ) t= 1 (5.33 ) Q = average length of queue in vehicles, w = average waiting time in the queue, in unit time per vehicle, t = average time spent in the system (w + 1/,u), in unit time per vehicle, an d Other terms are as defined previously.

5 5.5 Queuing Theory and Traffic Flow Analysis 163 EXAMPLE 5.10 Assume that the park attendant in Examples 5.7 and 5.9 takes an average of 15 seconds t o distribute brochures, but the distribution time varies depending on whether park patron s have questions relating to park operating policies. Given an average arrival rate of 18 0 veh/h as in Example 5.9, compute the average length of queue (in vehicles), averag e waiting time in the queue, and average time spent in the system, assuming M/M/1 queuing. Using the average arrival rate, departure rate, and traffic intensity as determined in Exampl e 5.9, the average length of queue is (from Eq ) _ = 2.25 veh the average waiting time in the queue is (from Eq. 5.32) 3 w _ 4(4 3 ) = 0.75 min/ve h and the average time spent in the system is (from Eq ) t= = 1 min/ve h M/MIN Queuing A more general formulation of the M/M/1 queue is the M/M/N queue, N is th e total number of departure channels. M/M/N queuing is a reasonable assumption at tol l booths on turnpikes or at toll bridges, there is often more than one departur e channel available (more than one toll booth open). A parking lot is another example, with N being the number of parking stalls in the lot and the departure rate, p, being the exponentially distributed times of parking duration. M/M/N queuing is also frequently encountered in non-transportation applications such as checkout lines a t retail stores, security checks at airports, and so on. The following equations describe the operational characteristics of M/M/N queuing. Note that unlike the equations for M/D/1 and M/M/1, which require that the traffic intensity, p, be less than 1, the following equations allow p to be greater than 1 but apply only when p/n (which is called the utilization factor) is less than 1.