Applications of the M^{X}/\text{semi-Markov}/1 queue to road traffic

Applications of the queue to road traffic

Abhishek Abhishek Rudesindo Núñez Queija Korteweg-de Vries Institute for Mathematics, University of Amsterdam, Amsterdam, The Netherlands
22email: {Abhishek, nunezqueija}@uva.nlMarko Boon Department of Mathematics and Computer Science, Eindhoven University of Technology, Eindhoven, The Netherlands
44email: m.a.a.boon@tue.nl
   Marko Boon Abhishek Rudesindo Núñez Queija Korteweg-de Vries Institute for Mathematics, University of Amsterdam, Amsterdam, The Netherlands
22email: {Abhishek, nunezqueija}@uva.nlMarko Boon Department of Mathematics and Computer Science, Eindhoven University of Technology, Eindhoven, The Netherlands
44email: m.a.a.boon@tue.nl
   Rudesindo Núñez-Queija Abhishek Rudesindo Núñez Queija Korteweg-de Vries Institute for Mathematics, University of Amsterdam, Amsterdam, The Netherlands
22email: {Abhishek, nunezqueija}@uva.nlMarko Boon Department of Mathematics and Computer Science, Eindhoven University of Technology, Eindhoven, The Netherlands
44email: m.a.a.boon@tue.nl
July 2, 2019
Abstract

The single server queue with multiple customer types and semi-Markovian service times, sometimes referred to as the queue, has been well-studied since its introduction by Neuts in 1966. In this paper, we apply an extension of this model, with batch arrivals and exceptional first service, to road traffic situations involving multiple streams of conflicting traffic. In particular, we use it in the context of gap acceptance models where low-priority traffic needs to cross (or, depending on the application, merge with) another traffic flow of higher priority.

Traditionally, gap acceptance models are based on the queue with exceptional first service, in this application area commonly referred to as the queue. In an earlier study AbhishekMergingModel (), we showed how the queue with exceptional first service can be applied in this context to extend the model with driver impatience and more realistic merging behaviour. In this paper, we show how this same queueing system can be used to model a Markov modulated Poisson arrival process of the high-priority traffic stream. Due to its flexibility, this arrival process is very relevant in this application, particularly because it allows the modelling of platoon forming of vehicles. The correlated inter-arrival times of these high priority vehicles cause the merging times of two subsequent low priority vehicles to become dependent as well (as they correspond with the service times in the underlying queueing model). We derive the waiting time and sojourn time distributions of an arbitrary customer, showing that these depend on the position of the customer inside the batch, as well as on the type of the first customer in the batch.

Keywords:
batch arrivals, queue, correlated service times, waiting time, sojourn time, gap acceptance models, Markov modulated Poisson process, unsignalized road intersections.

1 Introduction

The single server queue with multiple customer types and semi-Markovian service times, sometimes referred to as the queue, has been well-studied since its introduction by Neuts neuts66 (). An overview of the earlier existing literature AbhishekHeavyTraffic (); cin_s (); gaver (); neuts77a (); neuts77b () can be found in QUESTA2017 (), in which the transient and stationary queue length distributions in a single server model with batch arrivals and semi-Markov service times were analyzed. In this paper, we apply an extension of this model, with batch arrivals and exceptional first service, to road traffic situations involving multiple streams of conflicting traffic drew1 (); drew2 (); drew3 (); wei (); wu2001 (). In particular, we use it in the context of gap acceptance models abhishekcomsnets2016 (); AbhishekMergingModel (); heid94 (); heid97 () where low-priority traffic needs to cross (or, depending on the application, merge with) another traffic flow of higher priority. Drivers in the low-priority traffic flow wait until a sufficiently large gap arises between two subsequent vehicles in the high-priority traffic flow. This minimal gap, which may be vehicle-specific, is commonly referred to as the critical headway denoted by .

In the gap acceptance literature, three variations of driver behavior are distinguished (cf. AbhishekMMPPandImpatience (); heid97 ()). With the first behavior type (referred to as B in this paper), all low-priority vehicle drivers require the same constant critical headway to merge with the high-priority stream of vehicles. In the second case (B), which is commonly referred to as inconsistent gap acceptance behavior, each low-priority driver samples a critical headway from a given distribution at each new attempt. Its natural counterpart is known as consistent gap acceptance behavior (B), where the low-priority driver samples a critical headway from a given distribution only for his first attempt and then uses the same value at his subsequent attempts.

Traditionally, gap acceptance models are based on the queue with exceptional first service (cf. welch (); yeo (); yeoweesakul ()), in this application area commonly referred to as the queue. The “service times” correspond to the time required to search for a sufficiently large gap and crossing the intersection or, depending on the application, merging with the high-priority traffic flow. In an attempt to make the standard gap acceptance model more realistic, AbhishekMergingModel (), we developed a general framework based on the queue with batch arrivals and exceptional first service (which we refer to as the queue; see QUESTA2017 (); AbhishekHeavyTraffic ()) and showed how this queueing model can be applied in this context to extend the standard gap acceptance model with driver impatience and more realistic merging behaviour. In the present paper, we show how to exploit the versatility of the queue to model a Markov modulated Poisson arrival process of the high-priority traffic stream. Due to its flexibility, this arrival process is very relevant in this application, particularly because it allows the modelling of platoon forming of vehicles. We refer the reader to AbhishekMMPPandImpatience () for a brief overview of the earlier existing literature relevant to it. The correlated inter-arrival times of these high-priority vehicles cause the merging times of two subsequent low priority vehicles to become dependent as well (as they correspond with the service times in the underlying queueing model).

The contributions of this paper are twofold. First, we show how to derive the waiting time and sojourn time distributions of an arbitrary customer for the queueing system with exceptional first service, showing that these depend on the position of the customer inside the batch, as well as on the type of the first customer in the batch. Second, we focus on the application of this queueing model to road traffic situations involving multiple conflicting traffic streams, where on the minor road, vehicles arrive in batches according to a Poisson process and the arrival process on the major road is a Markov modulated Poisson process. Based on numerical examples, we demonstrate the impact of the three types of the driver behavior (B, B and B), on the delay on the minor road. More specifically, we show that the expected waiting times for the all three behavior types depend not only on the mean batch size, but also on the full distribution of the batch sizes.

The remainder of this paper is organized as follows. In Section 2.1, we present the description of the queueing model. Using the results from AbhishekHeavyTraffic (), we obtain the LST (Laplace-Stieltjes transform) of the steady-state waiting time and sojourn time distributions of customers as well as batches in Section 2.2. In Section 3, we first give several applications in which the extended queueing model arises, and then study the application to road traffic situations involving multiple conflicting traffic streams. In Section 4, we present the numerical examples.

2 The queueing model

In this section, we first describe the queuing model. Subsequently, we use the results from our paper AbhishekHeavyTraffic () on the steady-state distribution of the queue length, to derive the waiting time and sojourn time distributions.

2.1 Model description

Customers arrive in batches at a single-server queuing system according to a Poisson process with intensity . The arriving batch size is denoted by the random variable , with probability generating function (PGF) , for (zero-sized batches are not allowed, i.e. ). Customers are served individually and the service process is considered as a semi-Markov (SM) process. In addition, we assume that the first customer in each busy period has a different service time distribution than regular customers served in the busy period such that, for ,

(2.1)
(2.2)

where is the type of the -th customer and is its service time, and is the number of customers in the system at the departure of the -th customer.

In particular, for , we define

(2.3)
(2.4)

To be consistent with the terminology used in the gap acceptance literature, we refer to this queueing system as the queue. In this section, for improved readability, we briefly sketch the proof in AbhishekHeavyTraffic () to obtain the PGF of the queue length distribution at departure times of customers, which will be used to derive the waiting time and sojourn time distributions in the next section.

The queue length distribution at departure times can be obtained using the following recurrence relation:

(2.5)

where is the number of arrivals during the service time of the -th customer, and is the size of the batch in which -th customer arrived, with PGF , for . The conditional PGFs of the queue length distribution at departure epochs are obtained by solving the following system of equations:

(2.6)

where , , , and hence the PGF of the queue length distribution at departure epochs is given by

(2.7)

As customers arrive in the system according to a batch Poisson process with rate , for , we obtain,

(2.8)
(2.9)

2.2 Waiting time and sojourn time

In this section, we shall determine the waiting time and sojourn time distributions of an arbitrary batch as well as an arbitrary customer, noticing that the waiting time and sojourn time of a customer depend on its position in the batch, as well as on the type of service of the first customer in its batch.

To determine the waiting times and sojourn times of customers, firstly, we modify our model in such a way that all customers in the same batch are served together as a super customer. Let and be the service time and the service type of the -th super customer respectively. Then, the LST of the conditional service time of a super customer is defined as, for ,

(2.10)
(2.11)

Now, we can obtain the LST of the conditional service time of a super customer in terms of the LST of the conditional service time of an individual customer as

(2.12)
(2.13)

where is a matrix of order , and is the th element of matrix for .

Let be the number of super customers in the queue at the departure of, and the beginning of service of the -th super customer respectively. We can derive the PGF of the number of super customers in the queue, in steady state, at the departure of a super customer by letting , and in Equation (2.6).

Therefore, now, we know the distribution of the number of super customers at the departure of the super customer. But, to determine the waiting time of a super customer, using the distributional form of Little’s law, we need to find the distribution of the number of super customers at the beginning of the service of a super customer.

We can write

This implies that

(2.14)

where

(2.15)

and

(2.16)

Let and be the waiting time and sojourn time of the -th super customer respectively. By the distributional form of Little’s law, we obtain

Letting , then yields

(2.17)
(2.18)
(2.19)

Subsequently, we obtain

(2.20)
(2.21)

where .

Finally, we obtain the waiting times and sojourn times of individual customers in the batches by conditioning on the position of the customer in the batch, and using the following relations:

  • the waiting time of the first customer in the batch is equal to the waiting time of the super customer,

  • the waiting time of the -th customer in the batch, for , is equal to the waiting time of the super customer plus the service times of the first customers in the batch,

  • the sojourn time of the -th customer in the batch, for , is equal to the waiting time of the -th customer,

  • the sojourn time of the last customer in the batch is equal to the sojourn time of the super customer.

Let and be the steady-state waiting time and sojourn time of the -th customer served in his batch, respectively. Using the aforementioned relations, we obtain

(2.22)
(2.23)
(2.24)

Now, we are interested in the probability of being the -th customer served in a batch. For that, we define the arriving batch-size probabilities as for . Therefore, the probability that an arbitrary customer arrives in a batch of size , is equal to (see Burke Burke ()). And hence, the probability of being the th customer served in a batch is given by

(2.25)

Hence, the steady-state waiting and sojourn time LST of an arbitrary customer are given by

(2.26)
(2.27)
Remark 1

In case that batches have a maximum size of, say, , we can still use Equations (2.23) and (2.24). However, we note that although we define for , there is in fact no -th customer in the batch. Still, we need to define to determine . Alternatively, one can use .

3 Applications to road traffic

The queueing model considered in this paper arises in several applications including logistics, production/inventory systems, computer and telecommunication networks. In this section, we focus on the application to road traffic situations involving multiple conflicting traffic streams. More specifically, we consider an unsignalized priority-controlled intersection used by two traffic streams, both of which wish to cross the intersection (see Fig. 1). There are two priorities: the car drivers on the major road have priority over the car drivers on the minor road (and hence do not experience any impact from the car drivers on the minor road). The low-priority car drivers, on the minor road, cross the intersection as soon as they come across a gap with duration larger than between two subsequent high-priority cars, commonly referred to as the critical headway. On the major road, we consider Markov platooning (see also AbhishekMMPPandImpatience ()) which can be used to model the fluctuations in the traffic density with a dependency between successive gap sizes.

Figure 1: An example of an unsignalized intersection considered in this paper.

On the minor road, cars arrive in batches of size , with PGF , according to a Poisson process with rate . The arrival process on the major road is a Markov modulated Poisson process (MMPP) such that, for , is the Poisson rate when the continuous time Markov process (so-called background process), , is in phase . By introducing Markov platooning, an arrival process based on Markov modulation, we create a new, refined way of bunching on the major road. The semi-Markovian service times allow us to capture the required dependence between successive gap sizes. Platoon forming is a phenomenon that is frequently encountered in practice. Wu wu2001 () distinguishes between four different traffic flow regimes: free space (no vehicles), free flow (single vehicles), bunched traffic (platoons of vehicles), and queueing. In modern traffic manuals, it is suggested that intersection performance characteristics (such as capacity, which is the reciprocal of the mean service time) can be obtained by analyzing the intersection in one specific regime, and taking weighted averages of the steady-state performance measure under each of the regimes. However, this approach may lead to severe errors and it shown in AbhishekMMPPandImpatience () that one should build one model that captures all the variations in traffic flow instead. For this reason, we will show in this section how to use the single server queue with semi-Markovian service times to develop one gap acceptance model, capturing multiple traffic flow regimes on the major road by modeling them with a Markovian arrival process. We show how to obtain the service-time distributions of vehicles on the minor road for each of the three driver behavior types (B, B, and B), which can be plugged into the analysis of Section 2 to obtain the queue length PGF and waiting time LST. We will first define in more detail what we mean by service time in this application.

Definition 1 (Service time)

The service time of a vehicle on the minor road is the time between its arrival at the stop line and the moment when it has crossed the major road. The service time consists of two parts:

  • Scanning for a sufficiently large gap on the major road. This scanning time will be zero in case the remaining time until the next vehicle on the major road arrives is greater than the critical gap ;

  • Crossing the road, while freeing up the space for the next car to start scanning. This time is assumed to be equal to , i.e. exactly the size of the critical gap.

The transition probabilities of the background process of the MMPP are given by

with transition rate matrix

where .

Let and be the phase on the major road, seen by the -th low priority car at the beginning of its service when the -th car left the system non empty and empty respectively. In other words, we can say that is the phase on the major road when the -th car has crossed the major road. We can write

where which is given by


This implies that

We can write this in matrix form as

and hence we obtain

Now we determine the LST of the service time distribution for each of the three types of driver behavior.

3.1 (Constant critical gap)

Every driver on the minor road needs the same constant critical headway to enter the major road. Denote by the service time of the -th minor road car and the phase seen by the low priority car driver on the major road at time . We define, for ,

(3.1)
(3.2)

Now, firstly, we determine the probability that there is no car on the major road in and , given that . For that we define , with , and

(3.3)
(3.4)

We now present in Theorem 3.1 the service-time LST of vehicles on the minor road.

Theorem 3.1

The LST of the conditional service time for behavior type B is the solution to the following system of equations:

(3.5)

for .

Proof

First we solve the system of equations for by taking its Laplace-Stieltjes transform:

This implies that, for ,

(3.6)

We can write the above system of equations in matrix form as

where is a diagonal matrix and

After simplification, we obtain as

(3.7)

We readily find for , where is the inverse Laplace–Stieltjes transform operator.

Now, we need to determine the probability that at least one car arrives on the major road before time . Let be the time when the next car passes on the major road and . We will show that satisfies (3.4) by also taking its transform:

After simplification, we can write this as

and hence, in matrix form as

where .

Therefore, we obtain as

(3.8)

From Equations (3.7) and (3.8), we conclude the following relation

(3.9)

As a result, we obtain after taking the inverse Laplace-Stieltjes transform,

(3.10)

This leads to the conditional service-time LST

(3.11)

for , which can be rewritten to (3.5), proving the theorem.

Special case: Let , i.e., the MMPP is having two phases on the major road. In this case, we obtain from (3.7) as

(3.12)

Now, firstly, we determine the zeros (say ) of the polynomial which are given by

(3.13)

From Equation (3.13), we observe that the zeros and are real, distinct and non-positive. Moreover, without loss of generality, we assume that .

Therefore, we can write Equation (3.12) as

After partial fractions, we obtain

After taking the inverse Laplace transformation, the elements of the matrix are given by

(3.14)

From Equation (3.10), we obtain the following relations

(3.15)

Now, we know the expressions for and which we need to determine the LST of the conditional service time. For , Equation (3.5) becomes


For , after substituting the values of , we obtain the above expression as

After simplification, we can write this as