Traffic congestion at bus bays has decreased the service efficiency of public transit seriously in China, so it is crucial to systematically study its theory and methods. However, the existing studies lack theoretical model on computing efficiency. Therefore, the calculation models of bus delay at bays are studied. Firstly, the process that buses are delayed at bays is analyzed, and it was found that the delay can be divided into entering delay and exiting delay. Secondly, the queueing models of bus bays are formed, and the equilibrium distribution functions are proposed by applying the embedded Markov chain to the traditional model of queuing theory in the steady state; then the calculation models of entering delay are derived at bays. Thirdly, the exiting delay is studied by using the queueing theory and the gap acceptance theory. Finally, the proposed models are validated using fieldmeasured data, and then the influencing factors are discussed. With these models the delay is easily assessed knowing the characteristics of the dwell time distribution and traffic volume at the curb lane in different locations and different periods. It can provide basis for the efficiency evaluation of bus bays.
In recent years, with the rapid development of public transport, bus bays face an increasing pressure especially during peak hours. While serving passengers at a bus stop, buses can interact in ways that limit their discharge flows. This can increase bus delay at bays and degrade the bus system’s overall service quality [
Though professional handbooks [
The form of bus bay is shown in Figure
The behavior of queueing for serving at bus bays.
The berths are numbered 1 and 2 from the front to the back, and three buses arriving at the bus bay are numbered 1, 2, and 3 according to the arrival sequence. When buses 1 and 2 occupied berths 1 and 2 to serve their passengers, bus 3 must queue for entering upstream of the stop, as shown in Figure
In addition, when the serving is over at bus bays, the driver must look for a safe opportunity or “gap” in the traffic flow of curb lane to join them, as shown in Figure
The behavior of waiting for a gap at bus bays.
Therefore, the bus delays at bays mainly including entering delay and exiting delay are computed by the following equation:
We next derived the calculation models of bus entering delay and then studied the computing models of bus exiting delay. The calculation models of bus delay at singleberth and twoberth bays are proposed, respectively, finally.
At the bus bays serving some lines, buses enter the berth sequentially, then load and unload passengers, and finally exit the stop. So the buses and the stop constitute a queuing system [
Based on observations of bus operations in China, three assumptions are adopted in the course of formula derivation as follows.
It is assumed that bus overtaking maneuvers are prohibited, both within an entry queue and within the stop itself. Overtaking restrictions of this kind are common in cities, because an overtaking bus can disrupt car traffic in adjacent travel lanes.
The bus operating at bays is isolated from the effects of traffic signals.
The bus stop system operates in a stable state; the load rate
In this section, we firstly analyze and compute the transition probabilities of bus stop system; the balance equations are then formulated and solved for the Markov chain limiting probabilities; and, lastly, the models which are used to calculate the average bus entering delay are proposed.
Firstly, we define the Markov chain transition probabilities:
Let
Let
Let
According to (
State transition diagram of singleberth bays.
Let
In (
Equation (
So the expectation of
The variance of
Let
The equations are established according to the characteristics of the generating function, and then the balance equation of singleberth stop is resolved, as follows:
At the singleberth bus stop,
Combine (
The bus entering delay of twoberth bus bays is studied using the same approach as in Section
Let
For an
Then we determine the expression of each probability in (
(1)
(2)
We can derive the CDF of
Thus, for
(3)
In summary, the mathematical expectation of transition probabilities of twoberth bus bays is given by
The solution method of balance equation uses the
From the transition probabilities above, the balance equation of limiting probabilities can be written as
Then, we have
The
Let
Let
Hence,
Determining the average bus delay in queue requires the calculation of the average number of buses in queue over time. The average number of buses in queue is equal to the average of the queue length seen by each Poisson bus arrival. So it can be calculated by the next equation:
To obtain
No bus queues are present at the stop’s entry, both at the start and at the end of the
A bus queue is present at the start of cycle
A bus queue is present both at the start and at the end of cycle
A queue size greater than 2 is present at the start of cycle, and a queue thus persists at the end of that cycle; that is,
Note from the above that the
So,
Let
Therefore, we have
From Little’s formula [
The operability of (
According to the queueing theory and the gap acceptance theory, the average exiting delay is equal to the average number multiplied by the average length of nongaps that bus waits for, as shown in (
The probability that bus will be delayed is
The number of blocks is
The average number of vehicles between the starts of gaps is
Therefore, the average number of nongaps that bus waits for is
The total number of nongaps is
The average length of nongaps is
From this, it is noted that the average exiting delay is found by multiplying the average number by the average length of nongaps that bus waits for; that is,
Based on the above analysis, the average bus delay at singleberth bays is calculated by (
The average bus delay is calculated with (
The average bus delay at singleberth bays.
The average bus delay at twoberth bays is calculated by
The average bus delays are calculated with (
The average bus delay at twoberth bays.
The proposed model is validated using measured data at two bus bays of Tianmushan Road in Hangzhou city. The arriving time, queueing length, and service time of buses at Jingzhou North Intersection and Gudun Intersection bays during peak hours are surveyed by video. Then the delay of every bus is obtained by processing these data, as shown in Table
Comparison of the calculated and measured values of bus delay at bays.
Bus bay  Number of berths  Distribution of dwell time ( 
Bus flow (buses/h)  Car flow at curb lane (veh/h)  Calculated values (s)  Measured values (s)  Relative error (%) 

Jingzhou North Intersection  2  (2.856, 0.325)  96  360  5.68  5.12  9.86 
Gudun Intersection  2  (2.931, 0.414)  120  420  10.52  11.68  11.03 
Average  —  —  —  —  —  — 

The calculated results above show that the bus delay depends mainly on the average service time at given bus bays. The longer the average service time, the smaller the capacity of stop, which means the bus delay will increase. In addition, it is also affected by the coefficient of variation in bus service time at multiberth bus bays, and the impact characteristics are shown in Figure
Impact of coefficient of variation in bus service time on bus delay.
Formulas were developed to predict the average bus delay at bays. The formulas use a Markov chain that is embedded in the bus queueing process, the queueing theory, and the gap acceptance theory at these bays. Exact solutions were derived for two special cases: singleberth and twoberth bays. And approximations matched up to the surveyed results. With this methodology, the bus delays at bays are obtained easily if the characteristics of the service time distribution and traffic flow are known. And the results of this paper can provide basis for the efficiency evaluation of bus bays.
The authors declare that there is no conflict of interests regarding the publication of this paper.
This work is supported by the National Natural Science Foundation of China (no. 51278454) and the Shandong Province Natural Science Foundation (no. ZR2014EL036).