A quality of service–aware preemptive tidal flow queuing model for wireless multimedia sensor networks in the smart grid environment

Abstract

The smart grid incorporates a two-way communication system between customers and the utility for advanced monitoring and intelligent control of supply and demand. Wireless multimedia sensor network can be treated as an organic supplement and a peripheral network in this two-way communication system. However, the challenging smart grid environment makes it difficult to achieve a high quality of service in wireless multimedia sensor network. This article proposes a prioritization mechanism that considers the heterogeneous characteristics of smart grid traffic. Specifically, an innovative channel allocation and traffic scheduling scheme, named the preemptive tidal flow queuing model, is presented. This scheme achieves differentiated services for diverse communication data when the wireless multimedia sensor network accesses the core network and ensures the performance for high-priority data at the expense of the performance for low-priority data. Simulation analyses show that the performance for high-priority messages can be reliably guaranteed and that the preemptive tidal flow queuing model satisfies the requirements for a wireless multimedia sensor network operating in the smart grid environment. This article offers three main contributions: the development of a prioritization mechanism specifically for a wireless multimedia sensor network in the smart grid environment, the proposal of the preemptive tidal flow queuing model, and the presentation of formulas and simulations to verify the performance of the preemptive tidal flow queuing model.

Keywords

Smart grid wireless multimedia sensor network matrix theory prioritization mechanism preemptive tidal flow queuing model

Introduction

The smart grid is a new generation of power system in which various monitoring and actuating devices are employed. A smart grid autonomously monitors, diagnoses, controls, and efficiently operates the equipment used for power generation, distribution, and utilization.^1–3 As the essential link that enables intelligence in a smart grid, a two-way communication infrastructure is required to exchange real-time information between utilities and consumers.⁴

However, the question of how to simultaneously support a wide range of quality of service (QoS) requirements for numerous smart grid and legacy applications in this two-way communication infrastructure is an urgent problem to be solved. A typical smart grid communication infrastructure is illustrated in Figure 1. Because of their low costs for equipment and installation, rapid deployment, widespread access, and high flexibility, wireless communication network technologies play an extremely important role in the smart grid communication infrastructure,^5,6 and a wireless multimedia sensor network (WMSN), which can serve as an organic supplement and peripheral network in the two-way smart grid communication system, may be the most widely used wireless communication network technology for this purpose. The QoS-aware performance of a WMSN is suitable for smart grid applications, and the intelligent collection of data from many widely distributed sensors and actuators can be effectively realized. As shown in Figure 1, home area networks (HANs) and neighbor area networks (NANs) are the most suitable application scenarios for WMSNs. In HANs and NANs, a tremendous number of sensors and actuators are used for last-mile data collection, and enormous amounts of sensor data must be collected and processed. All data should be collected by smart meters (SMs) and aggregated in data aggregation units (DAUs) through the Meter Data Management System (MDMS) core access network. WMSN technology offers flexible and scalable aggregation and processing procedures for the large amounts of data collected in a many-sensor environment. The performance requirements (such as delay and reliability) for different types of data in the smart grid environment can be widely divergent, which can pose a serious challenge in wireless communication; therefore, the introduction of a specialized QoS scheme for smart grid WMSNs is extremely important.

Figure 1.

An overview of the smart grid communication infrastructure.

Numerous surveys have focused on the QoS problems faced in specific types of wireless communication networks, such as wireless mesh networks (WMNs), IEEE 802.11-based wireless local area networks (WLANs), and WMSNs. In the literature, QoS issues related to wireless sensor networks (WSNs) and WMSNs have been discussed widely. Azizi and Beghdad⁷ focused on maximizing the bandwidth reserved for each sensor node in a WSN and proposed the spiral-based clustered with data aggregation (SBCDA) architecture approach, which combines data aggregation with the time division multiple access (TDMA) protocol, for improving connectivity in WSNs and avoiding inter-cluster collisions.

Magaiaa et al.⁸ developed a new multi-objective approach to the WMSN routing problem that considers QoS requirements. Many recent studies have focused on energy efficiency, packet loss rate (PLR), and channel utilization efficiency.^9–11 However, the critical factor of transmission delay in smart grid WMSNs has not been fully considered in these papers. Chen¹² proposed a self-stabilizing hop-constrained energy-efficient (SHE) protocol for constructing minimum-energy networks for hard real-time routing; the transmission delay in a WSN was quantified, and the delay requirement was met, but this system does not provide a means of prioritizing messages and thus cannot satisfy the needs of smart grid WMSNs. Malik et al.¹³ analyzed the QoS features incorporated by the IEEE 802.11 standard and presented a systematic description of how to enhance the ability to provide high-QoS communication. As the authors stated, service differentiation and resource allocation play important roles in Internet QoS establishment, and similar ideas can be applied to WMSNs operating in the smart grid environment.

Regarding service differentiation, although standards such as IEEE 1646¹⁴ do provide qualitative requirements for different priority levels (such as high, medium, and low), the relative priorities of individual smart grid applications are not made specified. Similarly, IETF RFC 4594¹⁵ specifies qualitative priority requirements only for a few multimedia and enterprise applications. However, as described in Budka et al.,¹⁶ most network implementations support only three or four classes of services (other than the top class for network control traffic) to which the user classes of traffic are often mapped. Thus, traffic for many applications with diverse delay and priority requirements gets mapped to a single class and in return receives the same class-specific QoS treatment. In Deshpande et al.,¹⁷ priority values are assigned by the authors as their best estimates of the priorities of various applications based on the IEEE 1646 standard, IETF RFC 4594, and ITU-T Recommendations G.107¹⁸ and G.114,^19,20 combined with industry knowledge and engineering judgment. Priority ranks from 0 to 100 are assigned, with 0 indicating the highest priority and 100 indicating the lowest, which is a more suitable prioritization approach for the smart grid environment. In this article, the authors try to map more than one application to a QoS class and the corresponding queue in a smart grid network implementation. However, the authors fail to describe how to calculate the priorities of all applications.

Regarding resource allocation, there are two main architectural approaches that are used in wireless communication networks: integrated services (IntServ) and differentiated services (DiffServ). The DiffServ approach is typically the most suitable way to achieve a high QoS in a multi-priority system. In a typical DiffServ model, as analyzed in Demoor et al.,²¹ the high-priority packets in the communication system preempt all of the system’s resources, and all packets with lower priority must wait; however, when there are more than three levels of priority, the cascade affect will cause the performance of the medium-low level to be extremely poor, and the PLR for medium-low-priority messages cannot be guaranteed. Demoor et al.,²¹ Fan et al.,²² and Zhao et al.²³ made improvements to the DiffServ scheme for particular scenarios. Demoor et al.²¹ studied a two-class priority queue to model a DiffServ router with Expedited Forwarding Per-Hop Behavior for high-priority traffic, and the queue with finite capacity for the high-priority (class-1) packets has been studied in order to model a DiffServ router with Expedited Forwarding Per-Hop Behavior for class-1 packets. The presented model takes the exact class-1 queue capacity into account allowing the determination of class-1 packet loss and its influence on the performance of the system. And the intelligent queue management algorithms improve system performance when the handled traffic exhibits redundancy. In Fan et al.,²² an integrated routing risk model is constructed, which takes into account the effects of unicast routing on DiffServ network risk consisting of the impacts of interrupted services on network users and path availability. With the objective of minimizing integrated routing risk, a novel controllable chaotic immune routing algorithm (CCIRA) is proposed. And a path generation method based on chaotic search and dynamic adjacency matrix is proposed, improving the generation efficiency of available solutions of routing optimization algorithms. By combining the integrated routing risk model and CCIRA, the system can be highly efficient and practical, and the performance of this routing algorithm is proved to be superior. Zhao et al.²³ presented a service-oriented multi-domain control scheme in an attempt to achieve multi-domain quality of transmission (QoT) and energy-aware routing and spectrum assignment (RSA) in DiffServ networks, thereby striking a better balance between QoT and energy efficiency and also improving the utilization of spectrum resources with an acceptable average setup delay. However, no DiffServ communication network cannot be used directly in a smart grid WMSN because of the characteristic complexity of the smart grid environment, as the QoS requirements in smart grid are more detailed and the communication environment is more diverse.

Although many of the technologies discussed above are broadly applicable to the issue of QoS in WMSNs, smart grid WMSNs present several unique challenges that motivate the development of new methods because of the complex difficulties that arise in the smart grid environment, such as dynamic changes in topology, connectivity problems, interference, and fading. The existing prioritization mechanisms can provide only a small degree of fixed priority partitioning, which is not suitable for the smart grid environment, and the traditional channel allocation and traffic scheduling schemes are not designed for the typical operation conditions of smart grid WMSNs. To improve the QoS of WMSNs operating in the smart grid environment, we propose a prioritization mechanism that is designed specifically for this scenario. Based on this mechanism, an innovative channel allocation and traffic scheduling scheme called the preemptive tidal flow queuing model (PTFQM) is developed. Formulas are derived and performance simulations are presented to prove that the proposed prioritization mechanism is suitable for the smart grid environment and that the PTFQM can ensure the QoS of high-priority data in terms of delay and PLR.

The key contributions of this work are as follows:

A suitable prioritization mechanism is proposed to partition large numbers of messages into their appropriate priority levels. The priority partitioning is flexible and adjustable; the number of QoS levels can be customized by the user or automatically generated according to the scenario.

An innovative queuing model is proposed, the inspiration for which is the concept of tidal flow (or reversible) lanes for road traffic. In this queuing model, the channel resources for one QoS level can be diverted to messages of another QoS level in the case of congestion in a higher-level channel; in this way, the QoS for high-priority messages can be reliably guaranteed.

A simple calculation method for two-dimensional finite-length Markov chains is proposed. The method is based on Laplace transforms and the “matrix method,” and the solutions to the equations can be acquired by solving a matrix, which is a simple and intuitive approach.

Simulation results show that the PTFQM achieves efficient service differentiation while treating all QoS levels fairly under various conditions.

The remainder of the article is organized as follows: in section “Prioritization mechanism,” we discuss the architecture of our prioritization mechanism. In section “PTFQM,” we first explain the source of inspiration for our queuing model, we then provide an overall description of the PTFQM concept, and finally, we use pseudocode and flowcharts to clarify the specific steps of the PTFQM. In section “Performance evaluation,” MATLAB simulations are presented to demonstrate the performance improvement achieved using our prioritization mechanism and the PTFQM. Finally, we present concluding remarks in section “Conclusion.”

Prioritization mechanism

As an area of research involving highly intelligent systems, a specialized prioritization mechanism for smart grid WMSNs is urgently needed to adapt to the corresponding application requirements and provide a basis for a priority queuing model. However, as previously mentioned, the traditional standards can provide only simple priority divisions and fail to satisfy the requirements of individual smart grid applications. Therefore, to address this need, we propose our innovative prioritization mechanism. The proposed mechanism integrates power load ratings with message priorities; based on a QoS measure, we can subdivide the QoS levels based on environmental needs, thereby making the mechanism better adapted for WMSNs in the smart grid environment. Based on insight from the literature,²⁴ the QoS levels are determined using both subjective and objective weights. This approach can automatically adapt to users’ preferences by means of subjective weight calculations and allows service performance requirements to be accurately evaluated through objective weight calculations. With the application of this approach to both message priorities and final priorities, the method enables a more reasonable prioritization mechanism that is able to balance both power load and message requirements. In Zhao et al.,²³ the authors proposed a user-preference-adaptive subjective weight determination method (SWDM) and a service-potential-protective objective weight determination method (OWDM). The SWDM attempts to quantify user preferences and divide them into primary preferences and secondary preferences, based on certain restrictions, by calculating subjective weights related to individual preferences. The OWDM attempts to correct the one-sidedness of the priorities determined based on user preferences to achieve the best possible performance of the final results. This idea can also be applied in our prioritization mechanism using the SWDM to calculate the weight of the power load and using the OWDM to calculate the weights of a series of message attributes.

The system considered in this article is characterized as an information system

I = (WS, A)

(1)

where I represents the information system, which, in this article, is the WMSN communication system in the smart grid environment; WS represents the set of services, which are the differentiated services in the WMSN; and A represents the set of attributes, which include the power load and all communication attributes. The power load, denoted by $p_{0}$ , is included in the set of preferred attributes $P$ , and all communication attributes (such as traffic delay, bandwidth, and reliability), denoted by $p_{1}, p_{2}, \dots, p_{n}$ , are included in set $A$ ; thus, we have $P = {p_{0}}$ and $A = {p_{0}, p_{1}, p_{2}, \dots, p_{n}}$ , such that $P \subseteq A$ .

An attribute value $v_{ws} (p)$ is determined by users in accordance with the environment and the application requirements. Formulas (2) and (3) are then used to obtain the normalized attribute value $n v_{ws} (p)$

n v_{ws} (p) = {\begin{matrix} \frac{v_{ws} (p) - v_{p}^{min}}{v_{p}^{max} - v_{p}^{min}} & if v_{p}^{max} \neq v_{p}^{min} \\ 1 & if v_{p}^{max} = v_{p}^{min} \end{matrix}

(2)

n v_{ws} (p) = {\begin{matrix} \frac{v_{p}^{max} - v_{ws} (p)}{v_{p}^{max} - v_{p}^{min}} & if v_{p}^{max} \neq v_{p}^{min} \\ 1 & if v_{p}^{max} = v_{p}^{min} \end{matrix}

(3)

where $v_{p}^{max} = max {v_{ws} (p_{0}) | ws \in WS}$

and $v_{p}^{min} = min {v_{ws} (p_{0}) | ws \in WS}$ .

Subjective QoS measure

The subjective measure is used to capture user preferences. As summarized in Table 1, the ratings for power loads are clearly defined in GB50052-95. Power loads are distinguished based on the requirements with regard to power supply reliability and the impact of power supply interruption. Based on these power load ratings, we can assign attribute values $v_{ws} (p_{0})$ in descending order as shown in Table 1; higher levels are assigned larger attribute values.

Table 1.

Power load ratings.

Power load level	Characterization	Attribute value
Special grade	Particularly important location of level 1, power supply cannot be interrupted	4
Level 1	Interruption in the power supply will result in personal injury or death, cause a major political or economic loss, or have a major influence on significant political or economic organizations	3
Level 2	Interruption in the power supply will cause great political or economic damage or have an effect on the operation of important power units	2
Level 3	Any other power load	1

Because the power load is the only attribute in the preferred attribute set $P$ , we can directly obtain the subjective weight $sw (p) = 1$ .

Proceeding a step further, the subjective QoS measure can be derived as

\begin{matrix} SQ (ws) & = \sum_{p \in P} n v_{ws} (p) \cdot sw (p) = \sum_{p \in P} n v_{ws} (p) \\ = n v_{ws} (p_{0}) \end{matrix}

(4)

where $n v_{ws} (p_{0})$ is calculated using formulas (2) and (3).

Objective QoS measure

In our method, objective weights are applied to both the power load and the message attributes. In the algorithm presented in Ma et al.,²⁴ objective weights are calculated based on rough set theory. This implementation of rough set theory provides an objective form of analysis and thus is superior to classical rough set theory. Definitions 1–4 are adapted from Miao and Fan²⁵ and can be used to derive the reference level $REF (p)$ and the objective weight $ow (p)$ .

Definition 1. Consider an information system $I = (U, A, V, f)$ . Here, $U$ is a non-empty set of finite objects; $A$ is a non-empty, finite set of attributes; $V = \cup_{a \in A} V_{a}$ such that for $a \in A$ , $V_{a}$ is the set of values that attribute $a$ may take; and $f : U \times A \to V$ is the information function and is defined such that for any $a \in A$ and $u \in U$ , we have $f (u, a) \in V_{a}$ , $f (u, a)$ can be abbreviated as $a (u)$ , and $I$ can be abbreviated as $I = (U, A)$ .

Definition 2. Given an information system $I = (U, A)$ , for any $X \subseteq A$ , the indiscernibility relation $I_{x}$ is defined as

I_{x} = {(x, y) \in U \times U | \forall a \in X, f (x, a) = f (y, a)}

If $(x, y) \in I_{x}$ , then $x$ and $y$ are indiscernible in $X$ , which means that they have the same attribute value. Relation $I_{x}$ can also be called an associated equivalence relation or a knowledge relation. Moreover, the division determined by $X$ can be expressed as $U / I_{x}$ , or simply $U / X$ , which denotes the set of all equivalence classes divided by $I_{x}$ .

Definition 3. Given an information system $I = (U, A)$ , for $X \subseteq A$ , the granularity dimension of knowledge relation $I_{x}$ is defined as

GD (I_{x}) = | I_{x} | / | U^{2} | = | I_{x} | / {| U |}^{2}

where $| E |$ represents the number of elements in set $E$ and $GD (I_{x})$ can be abbreviated as $GD (x)$ ; under normal circumstances, $1 / | U | \leq GD (X) \leq 1$ . When $GD (x)$ is smaller, its distinguishing ability is stronger, and $I_{x}$ can be used to divide $U$ into more equivalence classes.

Definition 4. Given an information system $I = (U, A)$ , if $K \subseteq A$ , $κ \in A$ , and $I_{κ}$ is an indiscernibility relation in $K$ , then the significance of $κ$ with respect to $K$ is defined as

SI G_{K} (κ) = \frac{GD (K) - GD (K \cup {κ})}{GD (K)} = 1 - \frac{| I_{K \cup {κ}} |}{| I_{K} |}

This means that when $κ$ is added into $K$ , the attribute set $K$ can be used to distinguish more equivalence classes from $U$ . Under normal circumstances, $0 \leq SI G_{K} (x) \leq 1 - | I_{{κ}} | / {| U |}^{2}$ ; $κ$ becomes more important to $K$ as $SI G_{K} (κ)$ increases. $SI G_{\emptyset} (κ)$ represents the power of $κ$ to distinguish among different objects in the discourse domain $U$ .

Based on definitions 1–4, in the set of services $S = (WS, A, P)$ , $\forall p \in A$ , the reference level $REF (p)$ can be defined as

REF (p) = SI G_{\emptyset} (p) = 1 - | I_{{p}} | / {| WS |}^{2}

which reflects the distinguishing ability of $p$ toward services in the set $WS$ . In the same set $S$ , the objective weight $ow (p)$ for attribute $p (\in A)$ can be defined as

ow (p) = REF (p) / \sum_{p \in A} REF (p)

Furthermore, we can calculate the objective QoS measure of service $ws$ as follows

OQ (ws) = \sum_{p \in A} n v_{ws} (p) \cdot ow (p)

where $n v_{ws} (p)$ is the normalized attribute value, which can be derived from formulas (2) and (3).

QoS utility function

The QoS measure used in this article combines the subjective QoS measure $SQ (ws)$ and the objective QoS measure $OQ (ws)$ . The balance between the subjective and objective components is determined by a parameter $α$ , where $α \in [0, 1]$ ; this parameter is called the preference intensity and indicates the level of dependence on user preference. The QoS measure is calculated using the following QoS utility function

Q (ws) = α \cdot SQ (ws) + (1 - α) \cdot OQ (ws)

(5)

In this article, $α$ is defined by users, and we can achieve a balance between the power load and message requirements by adjusting $α$ . As $α$ increases, the effect of the power load becomes greater. Based on $Q (ws)$ , we can calculate QoS measures for all kinds of services, thereby providing a quantitative basis on which to divide different WMSN communication services with diverse power loads in every kind of application environment into different QoS levels.

The density of the QoS levels can be determined by users based on the smoothness requirements for message classification. The source node needs to use this method to calculate the priority of the packet before sending it, and the calculation in this section can be the pre-processing of our QoS-aware queuing model. The next section presents the proposed innovative PTFQM queuing model, which uses this QoS level calculation as its standard for prioritization and is therefore capable of providing different services adapted to all QoS levels, thereby achieving differential information processing. For easy calculation, we set the QoS level as three, in the actual scene, the density of the QoS levels can be greater.

PTFQM

Model description and analysis

The PTFQM, the innovative model proposed in this article, is based on the QoS levels of the numerous sensor data exchanged in a smart grid WMSN, which can be determined using the prioritization mechanism presented in the previous chapter. This innovative model is applied when data are gathered from sensor nodes into an SM or are uploaded from SMs to a DAU. When such a large amount of sensor data converges on an SM or a DAU, there is an unavoidable bottleneck in the WMSN’s access to the core network, which is also the key to enhancing the QoS of the entire communication network. The PTFQM attempts to process multiple levels of sensor data at this bottleneck in accordance with their priorities to supply superior service for high-priority data at the expense of the performance for low-priority data.

The inspiration for the PTFQM comes from the concept of tidal flow lanes for road traffic. Because of the “tide phenomenon” observed in traffic, some sections of a road will experience congestion in only a single direction at a time, as the peaks in traffic in opposite directions occur at different times. In this situation, an approach called the tidal flow system can be adopted for traffic diversion in these road segments. When one direction of the road is congested, the control center can change a free lane in the opposite direction into a temporary lane in the congested direction by switching the traffic lights to inform vehicles of the change; this lane is the tidal lane. The different lanes on a road are analogous to the channel resources that carry messages of different QoS levels when data are gathered from sensor nodes into an SM or are uploaded from SMs to a DAU. The messages being transmitted from the WMSN to the core network can be regarded as vehicles on a road, and vehicles traveling in different directions can be regarded as messages of different QoS levels, which traditionally should not occupy each other’s channel resources. However, if a tidal flow lane is established, then the appointed lane can be used for vehicles traveling in the other direction during a one-way peak period to ease the corresponding one-way congestion. Similarly, in the PTFQM, messages of one QoS level can occupy channel resources that are usually allocated to another QoS level in the case of congestion.

The PTFQM can be described as shown in Figure 2. Suppose that there are a total of $n$ QoS levels and that each QoS level has its own corresponding channel resources. As the index value increases, the priority of the QoS level decreases. The arrival rates for each level are mutually independent, and the messages of all $n$ levels follow Poisson processes with mean arrival rates of ${λ_{1}, λ_{2}, λ_{3}, \dots, λ_{n}}$ . All messages are served following the principle of first come, first served (FCFS), and messages of all levels are served according to exponential distributions with service rates of ${μ_{1}, μ_{2}, μ_{3}, \dots, μ_{n}}$ ; different levels of messages should be served at the same rate even in different channels. The $n$ QoS levels for messages are associated with $n$ channels, and the $j$ th channel contains $(j - 1)$ queues for all higher QoS levels (one-to-one mapping) and one queue for its own data; therefore, the $j$ th channel can be regarded as a single-server finite queuing model with $j$ levels of messages. In each channel, messages of higher priority are served before messages of lower priority, and after serving the last high-priority message present in the queue, the server automatically begins to serve the messages of the next level of priority that are waiting for service. The queue lengths for different levels of messages in the different channels can be adjusted according to user demand. As shown in Figure 2, the number of messages of level $i$ in the $j$ th channel is restricted to a finite number $L_{j, i} (1 \leq j \leq n, 1 \leq i \leq j)$ . A very important assumption in this model is that if a message overflows from a channel with a level lower than its own QoS level, then the delay of this message can already be regarded as beyond the tolerance limit, which means that the message can be immediately discarded, and the PLR can be calculated based on this understanding. By varying the queue lengths $L_{i, j}$ in the different channels, the performance of the queuing system can be modified; in this way, the optimal combination can be selected to ensure the optimal performance of the queuing model. $P_{oi} (i = 1, 2, 3, \dots, n - 1)$ represents the probability that the queue for messages of level $i$ in the corresponding channel $i$ is congested; in this case, any messages of level $i$ that subsequently arrive will overflow and divert channel resources from the lower QoS levels. Because the queue lengths in the $n$ th channel are set to infinity, there will be no overflow messages from the level $n$ queues. $τ_{i, j} (1 \leq i \leq n - 1, i < j \leq n)$ represents the probability that the next overflow message of level $i$ will preempt channel resources from level $j$ (one of the lower channels). The value of $τ_{i, j}$ can be determined using formula (6), from which we can infer that $τ_{i, j}$ is related to three key parameters: the priority of level $j$ (a higher value of $j$ indicates that the QoS level is lower and the corresponding channel is more suitable for preemption, which is reflected by an increase in $τ_{i, j}$ ), the message arrival rate for level $j$ (a higher value of $λ_{j}$ indicates that the channel is busier and should be preempted less frequently, which is reflected by a decrease in $τ_{i, j}$ ), and the queue length for messages of level $i$ on channel $j$ (a larger value of $L_{j, i}$ indicates that the ability of channel $j$ to carry data of level $i$ is stronger, which is reflected by an increase in $τ_{i, j}$ ; if $L_{j, i}$ is infinite, we set $L_{j, i} = 2 \cdot max [L_{j, i}, j \in (i, n - 1)]$ ). Moreover, the formula is subject to the following constraint conditions: $α_{1} - α_{2} + α_{3} = 1$ and $\sum_{μ = i + 1}^{n} τ_{i, μ} = 1 (i = 1, 2, 3, \dots, n - 1)$

\begin{matrix} \begin{matrix} τ_{i, j} = α_{1} \frac{j}{\sum_{μ = i + 1}^{n} μ} - α_{2} \frac{λ_{j}}{\sum_{μ = i + 1}^{n} λ_{μ}} \\ + α_{3} \frac{L_{j, i}}{\sum_{μ = i + 1}^{n} L_{μ, i}} \end{matrix} \end{matrix}

(6)

Figure 2.

A schematic diagram of the preemptive tidal flow queuing model.

The notation used to formulate the model is summarized below:

$n$ : number of QoS levels.

$λ_{i}$ $(i = 1, 2, \dots, n)$ : arrival rate for level $i$ messages.

$μ_{i}$ $(i = 1, 2, \dots, n)$ : service rate for level $i$ messages.

$L_{j, i} (1 \leq j \leq n, 1 \leq i \leq j)$ : upper limit on the number of level $i$ messages on channel $j$ .

$P_{oi} (i = 1, 2, \dots, n - 1)$ : probability that the queue for level $i$ messages in the corresponding channel $i$ will be congested.

$τ_{i, j} (1 \leq i \leq n - 1, i < j \leq n)$ : probability that the next overflow message of level $i$ will preempt channel resources of level $j$ .

$α_{1}$ , $α_{2}$ , $α_{3}$ : weights of the three key parameters used to calculate $τ_{i, j}$ .

Mathematical formulation of the model

The PTFQM can be regarded as a Markov model. To solve the system, we first develop a mathematical model for it. To reduce the computational complexity, the mathematical model is structured for the case of a three-priority architecture, that is, $n = 3$ .

Channel 1 system

In this model, level $1$ messages have the highest priority. Therefore, the channel $1$ system can be simplified as an $M / M / 1 / L_{1, 1}$ single-server finite waiting room queue system, in which the arrival and departure rates are not constant. Messages from sensor nodes arrive at a rate of $λ_{1}$ as long as there are fewer than $L_{1, 1}$ messages in the system, either in service or in the queue. When the waiting room becomes full, the next arriving message will be turned away. This may be expressed as follows

λ_{l} = {\begin{matrix} \begin{matrix} λ_{1} & l < L_{1, 1} \end{matrix} \\ \begin{matrix} 0 & l \geq L_{1, 1} \end{matrix} \end{matrix}

Because the service times for customers being served are exponentially distributed, the departure rate remains at $μ_{n} = μ_{1}$ for all $n$ .

In this case, the steady-state distribution of the number of messages in the system and the normalizing condition can be computed through direct use of the formulas for an $M / M / 1 / L_{1, 1}$ system.

The probability that a message will arrive at a full waiting room, $P_{o 1}$ , is given by

P_{o 1} = \frac{(1 - ρ) ρ^{L_{1, 1}}}{(1 - ρ^{L_{1, 1} + 1})}

(7)

Using the generating function for the number of messages in the system, we can determine the average number of messages in the waiting room queue to be

\begin{matrix} \bar{Q_{w}} = \sum_{k = 0}^{L_{1, 1} - 1} k P_{k + 1} = ρ^{2} P_{0} \sum_{k = 1}^{L_{1, 1} - 1} k ρ^{k - 1} = \frac{ρ}{1 - ρ} - \frac{L_{1, 1} ρ^{L_{1, 1} + 1} + ρ}{1 - ρ^{L_{1, 1} + 1}} \end{matrix}

(8)

The average number of messages in the server can be expressed as

\bar{Q_{s}} = 1 \times (1 - P_{0}) + 0 \times P_{0} = \frac{ρ - ρ^{L_{1, 1} + 1}}{1 - ρ^{L_{1, 1} + 1}}

(9)

Therefore, the average number of messages in the system is

{\bar{Q}}_{1, 1} = \bar{Q_{w}} + \bar{Q_{s}} = \frac{ρ}{1 - ρ} - \frac{(1 + L_{1, 1}) ρ^{L_{1, 1} + 1}}{1 - ρ^{L_{1, 1} + 1}}

(10)

We can use Little’s formula to calculate the average delay for level $1$ messages on channel $1$

\begin{matrix} {\bar{D}}_{1, 1} = \frac{\bar{Q}}{λ_{1}} = \frac{1 - (L_{1, 1} + 1) ρ^{L_{1, 1}} + L_{1, 1} ρ^{L_{1, 1} + 1}}{μ_{1} (1 - ρ) (1 - ρ^{L_{1, 1}})} \\ = \frac{1}{μ_{1} (1 - ρ)} - \frac{L_{1, 1} ρ^{L_{1, 1}}}{μ_{1} (1 - ρ^{L_{1, 1}})} \end{matrix}

(11)

Channel 2 system

Channel $2$ can be regarded as a single-server finite queuing model with two types of messages, level $1$ and level $2$ messages, with double orbits. The arrival of level $2$ messages follows a Poisson process with a mean arrival rate of $λ_{2}$ , and the arrival rate of level $1$ messages is given by

λ_{1, 2} = λ_{1} \times P_{o 1} \times τ_{1, 2} = λ_{1} \times \frac{(1 - ρ) ρ^{L_{1, 1}}}{(1 - ρ^{L_{1, 1} + 1})} \times τ_{1, 2}

(12)

The Chapman–Kolmogorov equations for the different states of the model (see Figure 3) are constructed as shown below:

1. Inactive state

The inactive state corresponds to the idle state of the server when neither priority nor non-priority customers are present in the system. The equation in this case is

\begin{matrix} {P'}_{0, 0} (t) = - (λ_{1, 2} + λ_{2}) P_{0, 0} (t) + μ_{2} P_{0, 1} (t) + μ_{1} P_{1, 0} (t) \end{matrix}

(13)

2. Busy state

The busy state corresponds to the state of the server when it is busy serving messages. Depending on the number of customers present in the system, various cases can arise, as follows:

1. When there are no level $2$ messages on channel $2$

\begin{matrix} {P'}_{i, 0} (t) = & - (λ_{1, 2} + λ_{2} + μ_{1}) P_{i, 0} (t) \\ + λ_{1, 2} P_{i - 1, 0} (t) + μ_{1} P_{i + 1, 0} (t) \end{matrix}

(14)

\begin{matrix} {P'}_{L_{2, 1}, 0} (t) = & - (λ_{2} + μ_{1}) P_{L_{2, 1}, 0} (t) \\ + λ_{1, 2} P_{L_{2, 1} - 1, 0} (t) \end{matrix}

(15)

2. When both level $1$ and level $2$ messages are present on channel $2$

\begin{matrix} {P'}_{i, j} (t) = & - (λ_{1, 2} + λ_{2} + μ_{1}) P_{i, j} (t) + λ_{1, 2} P_{i - 1, j} (t) \\ + λ_{2} P_{i, j - 1} (t) + μ_{1} P_{i + 1, j} (t) \end{matrix}

(16)

\begin{matrix} {P'}_{L_{2, 1}, j} (t) = & - (λ_{2} + μ_{1}) P_{L_{2, 1}, j} (t) + λ_{1, 2} P_{L_{2, 1} - 1, j} (t) \\ + λ_{2} P_{L_{2, 1}, j - 1} (t) \end{matrix}

(17)

\begin{matrix} {P'}_{i, L_{2, 2}} (t) = & - (λ_{1, 2} + μ_{1}) P_{i, L_{2, 2}} (t) + λ_{1, 2} P_{i - 1, L_{2, 2}} (t) \\ + λ_{2} P_{i, L_{2, 2} - 1} (t) + μ_{1} P_{i + 1, L_{2, 2}} (t) \end{matrix}

(18)

\begin{matrix} {P'}_{L_{2, 1}, L_{2, 2}} (t) = & - μ_{1} P_{L_{2, 1}, L_{2, 2}} (t) + λ_{1, 2} P_{L_{2, 1} - 1, L_{2, 2}} (t) \\ + λ_{2} P_{L_{2, 1}, L_{2, 2} - 1} (t) \end{matrix}

(19)

3. When there are no level $1$ messages on channel $2$

\begin{matrix} {P'}_{0, j} (t) = & - (λ_{1, 2} + λ_{2} + μ_{2}) P_{0, j} (t) + λ_{2} P_{0, j - 1} (t) \\ + μ_{2} P_{0, j + 1} (t) + μ_{1} P_{1, j} (t) \end{matrix}

(20)

\begin{matrix} {P'}_{0, L_{2, 2}} (t) = & - (λ_{1, 2} + μ_{2}) P_{0, L_{2, 2}} (t) + λ_{2} P_{0, L_{2, 2} - 1} (t) \\ + μ_{1} P_{1, L_{2, 2}} (t) \end{matrix}

(21)

Figure 3.

The state transition diagram for the channel 2 system.

For all states presented above, we have the normalizing condition

\sum_{i = 0}^{L_{2, 1}} \sum_{j = 0}^{L_{2, 2}} P_{i, j} (t) = 1

(22)

The transient solution to this system of equations can be obtained using the “matrix method.” The matrix method is a well-established technique for determining the solution to a set of differential equations. Before proceeding further, we define the transient-state probabilities in terms of $π (t)$ as

\begin{matrix} Π_{1} (t) = [π_{1} (t), π_{2} (t), \dots, \\ {π_{L_{2, 2}} (t), π_{L_{2, 2} + 1} (t)]}_{L_{2, 2} + 1}^{T} \end{matrix}

(23)

\begin{matrix} Π_{l + 2} (t) = [π_{L_{2, 2} + 2 + (l \times L_{2, 1})} (t), π_{L_{2, 2} + 3 + (l \times L_{2, 1})} (t), \dots, \\ {π_{L_{2, 2} + (l + 1) \times L_{2, 1}} (t), π_{L_{2, 2} + 1 + (l + 1) \times L_{2, 1}} (t)]}_{L_{2, 1}}^{T} \end{matrix}

(24)

We denote the Laplace transforms of $π_{i} (t)$ , $Π_{i} (t)$ , and $Π (t)$ by ${\tilde{π}}_{i} (s)$ , ${\tilde{Π}}_{i} (s)$ , and $\tilde{Π} (s)$ , respectively. In terms of Laplace transforms, the above transient-state probabilities reduce to

\begin{matrix} \tilde{Π} (s) = [{\tilde{Π}}_{1} (s), {\tilde{Π}}_{2} (s), {\tilde{Π}}_{3} (s), \dots, {\tilde{Π}}_{L_{2, 2} + 1} (s), \\ {{\tilde{Π}}_{L_{2, 2} + 2} (s)]}_{(L_{2, 1} \times L_{2, 2} + L_{2, 1} + L_{2, 2} + 1) \times 1}^{T} \end{matrix}

(25)

Note that initially, at time $t = 0$ , the system is empty, that is,

$P_{0, 0} (0) = 1$ and $P_{i, j} (0) = 0 \forall i \neq 0, j \neq 0$ . The initial vector can be defined as

\begin{matrix} Π (0) = [1, 0, 0, \dots, 0, 0, 0, \dots, 0]_{(L_{2, 1} \times L_{2, 2} + L_{2, 1} + L_{2, 2} + 1) \times 1}^{T} \end{matrix}

(26)

As the first step, we take the Laplace transforms of the equations to convert them into a differential-free form. After Laplace transformation, the set of differential equations (15)–(23) can be rewritten in matrix form as

A (s) \tilde{Π} (s) = Π (0)

(27)

Here

A (s) = {(\begin{matrix} A_{0} & C_{0} & C_{1} & \dots & C_{l} & \dots & C_{L_{2, 2}} \\ B_{0} & D \\ B_{1} & F & D \\ ⋮ & F & ⋱ \\ B_{l} & ⋱ & ⋱ \\ ⋮ & ⋱ & D \\ B_{L_{2, 2}} & F & E \end{matrix})}_{M \times M},

(28)

where $M = (L_{2, 1} \times L_{2, 2} + L_{2, 1} + L_{2, 2} + 1)$ .

From equation (27), we obtain the following (cf. Jain et al.²⁶)

\begin{matrix} {\tilde{π}}_{i} (s) = \frac{det [A_{i} (s)]}{det [A (s)]} \\ (i = 1, 2, 3, \dots, (L_{2, 1} \times L_{2, 2} + L_{2, 1} + L_{2, 2} + 1)) \end{matrix}

(29)

where $det [A (s)]$ is the determinant of the matrix $A (s)$ and $det [A_{i} (s)]$ is the determinant of the matrix obtained by replacing the $i$ th column vector $(i = 1, 2, 3, \dots, (L_{2, 1} \times L_{2, 2} + L_{2, 1} + L_{2, 2} + 1))$ of $A (s)$ with the initial vector $Π (0)$ . We can write an explicit expression for equation (29) as follows

\begin{matrix} {\tilde{π}}_{i} (s) = \frac{a_{0}}{s} + \sum_{m = 1}^{x} \frac{a_{m}}{s + δ_{m}} \\ + \sum_{m = 1}^{y} \frac{b_{m} s + c_{m}}{s^{2} + (δ_{x + m} + {\bar{δ}}_{x + m}) s + δ_{x + m} {\bar{δ}}_{x + m}} \\ (i = 1, 2, 3, \dots, (L_{2, 1} \times L_{2, 2} + L_{2, 1} + L_{2, 2} + 1)) \end{matrix}

(30)

where

a_{0} = \frac{det [A_{i} (0)]}{(Π_{k = 1}^{x} δ_{k}) (Π_{k = 1}^{y} δ_{x + k} {\bar{δ}}_{x + k})}

(31)

\begin{matrix} a_{m} = det [A_{i} (- δ_{m})] / \\ {(- δ_{m}) [Π_{k = 1, k \neq m}^{x} (δ_{k} - δ_{m})] \times \\ [Π_{k = 1}^{y} {{(- δ_{m})}^{2} + (δ_{x + k} + {\bar{δ}}_{x + k}) (- δ_{m}) + δ_{x + k} {\bar{δ}}_{x + k}}]} \\ (m = 1, 2, \dots, x) \end{matrix}

(32)

\begin{array}{l} b_{m} (- δ_{x + m}) + c_{m} = \det [A_{i} (- δ_{x + m})] / \\ {(- δ_{x + m}) \times [\prod_{k = 1}^{x} (δ_{k} - δ_{x + m})] \times \\ [\prod_{k = 1, k \neq m}^{y} ({(- δ_{x + m})}^{2} + (δ_{x + k} + {\bar{δ}}_{x + k}) (- δ_{x + m}) + \\ δ_{x + k} {\bar{δ}}_{x + k})]} (m = 1, 2, \dots, y) \end{array}

(33)

Upon taking the inverse Laplace transform of equation (30), the probabilities of the system states at any time $t$ are given by

\begin{matrix} π_{i} (t) = a_{0} + \sum_{m = 1}^{x} a_{m} e^{- δ_{m} t} \\ + \sum_{m = 1}^{y} [b_{m} e^{- ν_{m} t} \cos (w_{m} t) + \frac{c_{m} - b_{m} ν_{m}}{w_{m}} e^{- ν_{m} t} \sin (w_{m} t)] \\ 1 \leq i \leq (L_{2, 1} \times L_{2, 2} + L_{2, 1} + L_{2, 2} + 1) \end{matrix}

(34)

where $a_{0}$ , $a_{m}$ , $b_{m}$ , $c_{m}$ , $ν_{m}$ , and $w_{m}$ are real numbers.

Let $n = ⌊ i / L_{2, 2} ⌋$ and $m = (i mod L_{2, 2})$ ; then, we can write $π_{i} (t) = P_{n, m} (t)$ . The probability of a level $2$ message arriving at a full waiting room, $P_{o 2}$ , is given by

P_{o 2} = \sum_{n = 0}^{L_{2, 1}} P_{n, L_{2, 2}} (t)

(35)

The expected number of level $1$ messages in the channel $2$ system at time $t$ is

{\bar{Q}}_{2, 1} = (1 - \sum_{m = 0}^{L_{2, 2}} P_{0, m} (t)) + \sum_{n = 1}^{L_{2, 1}} \sum_{m = 0}^{L_{2, 2}} n P_{n, m} (t)

(36)

and the expected number of level $2$ messages in the channel $2$ system at time $t$ is

\begin{matrix} {\bar{Q}}_{2, 2} = (1 - P_{0, 0} (t) - \sum_{n = 1}^{L_{2, 1}} \sum_{m = 0}^{L_{2, 2}} P_{n, m} (t)) \\ + \sum_{n = 0}^{L_{2, 1}} \sum_{m = 1}^{L_{2, 2}} m P_{n, m} (t) \end{matrix}

(37)

We can use Little’s formula to calculate the average delays for level $1$ and level $2$ messages, that is, ${\bar{D}}_{2, 1}$ and ${\bar{D}}_{2, 2}$ , respectively.

Channel 3 system

All three queues for messages of different levels on channel $3$ are regarded as being of infinite length. The arrival of level $3$ messages follows a Poisson process with a mean arrival rate of $λ_{3}$ . The arrival rate for level $2$ messages on channel $3$ can be derived as follows

λ_{2, 3} = λ_{1} P_{o 2} τ_{2, 3} = λ_{2} τ_{2, 3} \sum_{n = 0}^{L_{2, 1}} P_{n, m} (t)

Finally, level $1$ messages arrive on channel $3$ with an arrival rate of

λ_{1, 3} = λ_{1} P_{o 1} τ_{1, 3} = λ_{1} \frac{(1 - ρ) ρ^{L_{1, 1}}}{(1 - ρ^{L_{1, 1} + 1})} τ_{1, 3}

Because all three queues are infinite, we can refer to Gross and Harris²⁷ to find that in the case of $r$ priority levels and exponential service distributions, the moments of the numbers of messages in the system can be obtained as follows

{\bar{Q}}^{(i)} = \frac{ρ_{i}}{1 - σ_{i - 1}} + \frac{λ_{i} \sum_{j = 1}^{i} λ_{j} E [S_{j}^{2}]}{2 (1 - σ_{i - 1}) (1 - σ_{i})}

(38)

where $S_{j}$ is the service time for a level $j$ message; because we set all messages of the same level to be of a fixed length and because they are served according to an exponential distribution, $E [S_{i}]$ is a constant equal to $1 / μ_{i}$ , $E [S_{i}^{2}] = 2 / μ_{i}$ , $ρ_{i} = λ_{i} E [S_{i}] = λ_{i} / μ_{i}$ , and $σ_{i} = \sum_{j = 1}^{i} ρ_{i}$ . Therefore, we have

{\bar{Q}}_{3, 1} = \frac{ρ_{1}}{1 - ρ_{1}}

(39)

{\bar{Q}}_{3, 2} = \frac{ρ_{2} - ρ_{1} ρ_{2} + ρ_{1} ρ_{2} (μ_{2} / μ_{1})}{(1 - ρ_{1}) (1 - ρ_{1} - ρ_{2})}

(40)

{\bar{Q}}_{3, 3} = \frac{ρ_{3} - ρ_{1} ρ_{3} - ρ_{2} ρ_{3} + ρ_{1} ρ_{3} (μ_{3} / μ_{1}) + ρ_{2} ρ_{3} (μ_{3} / μ_{2})}{(1 - ρ_{1} - ρ_{2}) (1 - ρ_{1} - ρ_{2} - ρ_{3})}

(41)

where $ρ_{1} = λ_{1, 3} / μ_{1}$ , $ρ_{2} = λ_{2, 3} / μ_{2}$ , and $ρ_{3} = λ_{3} / μ_{3}$ .

Using the average queue lengths and arrival rates given above, we can obtain the average delays for messages of the different levels in the channel $3$ system (i.e. ${\bar{D}}_{3, 1}$ , ${\bar{D}}_{3, 2}$ , and ${\bar{D}}_{3, 3}$ ) through multiple applications of Little’s formula.

Performance analysis

The validity of the PTFQM can be best evaluated in terms of performance indices. Various indices, namely, average queue length, average delay, PLR, and throughput, can be used to judge the efficiency of the PTFQM. Some of these performance measures are as follows:

1. Average queue length

The average queue lengths in the different channel systems are elaborated in the previous sections and need not be reproduced here.

2. Average delay

The average delay for messages at level $i$ should be determined by fusing the delays for level $i$ messages in all channel systems. The formulas can be derived as follows:

Average delay for level $1$ messages in the PTFQM system

{\bar{D}}_{1} = λ_{1} (1 - P_{o 1}) {\bar{D}}_{1, 1} + λ_{1} P_{o 1} τ_{1, 2} {\bar{D}}_{2, 1} + λ_{1} P_{o 1} τ_{1, 3} {\bar{D}}_{3, 1}

(42)

Average delay for level $2$ messages in the PTFQM system

{\bar{D}}_{2} = λ_{2} (1 - P_{o 2}) {\bar{D}}_{2, 2} + λ_{2} P_{o 2} τ_{2, 3} {\bar{D}}_{3, 2}

(43)

Average delay for level $3$ messages in the PTFQM system

{\bar{D}}_{3} = {\bar{D}}_{3, 3}

(44)

3. PLR

In the three-priority PTFQM system, packet loss occurs only for level $1$ messages; if the number of level $1$ messages exceeds the queue length $L_{2, 1}$ in the channel $2$ system, then the overflow messages will be discarded. Thus, we have

PL R_{1} = \sum_{m = 0}^{L_{2, 2}} P_{L_{2, 1}, m} (t)

(45)

For an $i$ -priority system, messages of levels $1$ through $(i - 2)$ will be subject to packet loss, and the value of $PL R_{j}$ should be obtained by fusing the $PLR$ values for level $j$ messages on channels $(j + 1) ~ (i - 1)$ .

Performance evaluation

In this section, we present numerical and analog simulations of the PTFQM. Both the numerical experiments and the simulation model were implemented using MATLAB to verify the proposed computational approach. Moreover, a comparison between the PTFQM, the non-preemptive queuing model (NP model), and the DiffServ model is presented to demonstrate the applicability and dependability of the PTFQM for a WMSN in the smart grid communication environment. By adjusting various parameters in the PTFQM system, we can learn how these parameters affect the performance of the system.

Verification of the numerical simulation

The default parameter values used to verify the numerical simulation of the system are provided below:

$n = 3$ , $λ_{1} = 0.01 - 0.9$ , $λ_{2} = 0.85$ , $λ_{3} = 0.9$ , $μ_{1} = 0.95$ , $μ_{2} = 0.93$ , $μ_{3} = 0.91$ , $α_{1} = 5.85$ , $α_{2} = 2.45$ , $α_{3} = - 2.4$ , $L_{1, 1} = 5$ , $L_{2, 1} = 2$ , $L_{2, 2} = 5$ , and $L_{3, 1} = L_{3, 2} = L_{3, 3} = \infty$ .

Figure 4 depicts the average delay for level $1$ messages in a system with three priority levels. With $λ_{1}$ as the horizontal coordinate, the figure shows that the sampled points from the numerical simulation are essentially consistent with the curve obtained in the analog simulation, indicating that the formula derivation presented in the previous chapter is correct.

Figure 4.

Comparison between the results of numerical and analog simulations.

Performance comparison with the NP model and the DiffServ model

Traditional queuing models, such as the NP model, use independent channels for messages of different priorities; each channel carries messages only of the corresponding priority, and when the number of messages surpasses the queue length of the corresponding channel, any further packets will be discarded. Meanwhile, in the DiffServ model, as described in Ward et al.,²⁰ messages of any priority are served only if there are no messages of higher priority in the system, and the messages that are currently receiving service occupy all resources in the system; in other words, the system is a multi-priority single-server system, in which the messages of highest priority are serviced first and the FCFS principle is obeyed for messages at the same level. Figure 5 shows the results of a performance comparison between the PTFQM, the NP model, and the DiffServ model. The default parameter values in the PTFQM and the NP model are provided below:

$n = 5$ , $λ_{1} = 0.01 - 0.6$ , $λ_{2} = 0.65$ , $λ_{3} = 0.7$ , $λ_{4} = 0.8$ , $λ_{5} = 0.85$ , $μ_{1} = 0.97$ , $μ_{2} = 0.95$ , $μ_{3} = 0.93$ , $μ_{4} = 0.91$ , $μ_{5} = 0.9$ , $L_{1, 1} = 5$ , $L_{2, 1} = 2$ , $L_{3, 1} = 3$ , $L_{4, 1} = 4$ , $L_{2, 2} = 5$ , $L_{3, 2} = 3$ , $L_{4, 2} = 4$ , $L_{3, 3} = 5$ , $L_{4, 3} = 3$ , $L_{4, 4} = 5$ , and $L_{5, 1} = L_{5, 2} = L_{5, 3} = L_{5, 4} = L_{5, 5} = \infty$ .

Figure 5.

Performance comparison between the PTFQM and the NP model (Part I): (a) influence of $λ_{1}$ on ${\bar{D}}_{1}$ , (b) influence of $λ_{1}$ on $PL R_{1}$ , (c) influence of $λ_{1}$ on ${\bar{D}}_{2}$ , and (d) influence of $λ_{1}$ on $PL R_{2}$ .

The parameters of the DiffServ model are different with regard to channel resources and service rates:

$n = 5$ , $λ_{1} = 0.01 - 0.6$ , $λ_{2} = 0.65$ , $λ_{3} = 0.7$ , $λ_{4} = 0.8$ , $λ_{5} = 0.85$ , $μ = 4.5$ , $L_{1} = 5$ , $L_{2} = 5$ , $L_{3} = 5$ , $L_{4} = 5$ , and $L_{5} = \infty$ .

Figure 5(a) depicts the average delay for level $1$ messages as a function of the arrival rate of level $1$ messages, $λ_{1}$ . For all three models, the average delay increases with increasing $λ_{1}$ . It is obvious that in the PTFQM, level $1$ messages have a lower average delay than in the NP model, and as $λ_{1}$ increases, the advantage of the PTFQM becomes more significant; meanwhile, level $1$ messages in the DiffServ model can have the lowest average delay because in this model, level $1$ messages can monopolize all resources in the system. However, the average delays in all three models are minimal, and the slight increase in delay in the PTFQM compared with the DiffServ model is offset by the former’s advantage in terms of the $PLR$ .

Figure 5(b) presents the comparison between the PTFQM, the NP model, and the DiffServ model in terms of the $PLR$ for level $1$ messages as a function of $λ_{1}$ . The $PLR$ for level $1$ messages in the PTFQM is very small, almost negligible, whereas the $PLR$ in the NP model shows a significant, sustained increase when $λ_{1} > 0.3$ . Similarly, the $PLR$ for level $1$ messages in the DiffServ model also shows a sustained increase when $λ_{1} > 0.3$ , although the growth trend is slower than in the NP model. The $PLR$ for level $1$ messages in the PTFQM will increase to some extent with a finite $L_{2, 1}$ , $L_{3, 1}$ , and $L_{4, 1}$ , but this increase will be far less than that in the other two models.

Figure 5(c) depicts the average delay for level $2$ messages as a function of $λ_{1}$ . When $λ_{1} < 0.4$ , the value of ${\bar{D}}_{2}$ in the DiffServ model is less than in the other two models; however, ${\bar{D}}_{2}$ also increases faster as a function of $λ_{1}$ in the DiffServ model than in the other two models because the number of level $1$ messages affects ${\bar{D}}_{2}$ much more strongly in the DiffServ model. When $λ_{1} > 0.4$ , in the DiffServ model, the value of ${\bar{D}}_{2}$ rapidly increases with increasing $λ_{1}$ and exceeds the value in the PTFQM. This is because once $λ_{1}$ increases to a sufficient extent, the level $1$ messages will occupy all resources in the system for a sustained period of time, and the level $2$ messages in the system will be required to wait. With a further increase in $λ_{1}$ , the value of ${\bar{D}}_{2}$ in the DiffServ model eventually exceeds that in the NP model, as well. When $λ_{1} < 0.6$ , the value of ${\bar{D}}_{2}$ in the PTFQM is less than that in the NP model; however, when $λ_{1} > 0.6$ , the number of level $1$ messages becomes excessive, and overflowing level $1$ messages will frequently occupy resources from channels $2$ – $5$ , causing level $2$ messages to wait longer in their queues.

Figure 5(d) shows the $PLR$ values for level $2$ messages in the PTFQM, the NP model, and the DiffServ model. The $PLR$ for level $2$ messages in the PTFQM is equal to zero when $λ_{1} < 0.5$ ; when $λ_{1} > 0.5$ , the $PLR$ for level $2$ messages in the PTFQM shows a slight increase with increasing $λ_{1}$ , but it is still far less than in the other two models because the overflowing level $2$ messages can occupy the resources of lower priority channels. The $PLR$ for level $2$ messages in DiffServ is far less than that in the NP model; however, it increases more rapidly than in the PTFQM because overflowing level $2$ messages are directly discarded.

Figure 6(a) presents the average delay for level $3$ messages as a function of $λ_{1}$ . In the PTFQM, as the number of level $1$ messages becomes excessive, some of the overflowing level $1$ messages preempt resources from channel $3$ , and some of the overflowing level $2$ messages also preempt resources from channel $3$ , thereby leading to an increase in the average delay of level $3$ messages when $λ_{1} > 0.3$ . However, in the DiffServ model, the system resources will be entirely occupied by level $1$ and level $2$ messages for a long time, leading to a rapid increase with increasing $λ_{1}$ .

Figure 6.

Performance comparison between the PTFQM and the NP model (Part II): (a) influence of $λ_{1}$ on ${\bar{D}}_{3}$ , (b) influence of $λ_{1}$ on $PL R_{3}$ , (c) influence of $λ_{1}$ on ${\bar{D}}_{4}$ , and (d) influence of $λ_{1}$ on ${\bar{D}}_{5}$ .

Figure 6(b) depicts the $PLR$ values for level $3$ messages in all three models. It is obvious that the PTFQM has a much lower $PLR$ for level $3$ messages than the other two models do. This is because when the number of level $3$ messages in their corresponding queue exceeds the length of that queue, the NP model and the DiffServ model immediately discard the excess messages, whereas in the PTFQM, overflowing level $3$ messages can take over channel resources from level $4$ and level $5$ to resume transfer. In fact, the same situation arises with regard to the $PLR$ values for level $4$ messages in all three models; to avoid repetition, this scenario is not explicitly described in this article.

Figure 6(c) shows the average delay for level $4$ messages as a function of $λ_{1}$ for all three models. The values of ${\bar{D}}_{4}$ in the DiffServ model and the PTFQM are both larger than that in the NP model, and both increase with increasing $λ_{1}$ ; this is because level $4$ messages can be affected by high-priority messages in these two models. However, the value of ${\bar{D}}_{4}$ in the PTFQM is superior to that in the DiffServ model because high-priority messages are shared with other lower priority channels and overflowing level $4$ messages can also preempt channel resources from level $5$ .

Figure 6(d) shows the average delay for level $5$ messages as a function of $λ_{1}$ for all three models. The slowdown in the growth trend compared with high-priority messages is because level $5$ messages are affected by messages at all higher levels, and consequently, ${\bar{D}}_{5}$ can be very long in both the PTFQM and the DiffServ model, which causes an increase in $λ_{1}$ to have relatively little effect.

These comparisons between the PTFQM, the NP model, and the DiffServ model show that the performance of the PTFQM in terms of the average delay and $PLR$ for high-priority messages is obviously superior. Although the performance of the PTFQM falls short of that of the DiffServ model for ${\bar{D}}_{1}$ and ${\bar{D}}_{2}$ , the difference is minimal; the obvious advantage of the PTFQM in terms of the $PLR$ reflects a considerable improvement in reliability, and its advantage in terms of the average delays for medium- and low-priority messages indicates a certain improvement in fairness compared with the DiffServ model. Generally, the PTFQM can achieve differentiated services for diverse communication data in a WMSN operating in the smart grid environment by ensuring the performance for high-priority data at the expense of the performance for low-priority data while nevertheless maintaining better overall fairness than that in the DiffServ model, as manifested in the average delays for medium- and low-priority messages, and better overall reliability, as manifested by the $PLR$ values for all levels. When a smart grid WMSN communication system accesses the core network, the time available for acquiring common electricity data can be on the order of days, and such delays for low-priority messages are therefore acceptable, whereas the delays and $PLR$ values for high-priority messages in a smart grid system should be as small as possible. In summary, the innovative PTFQM proposed in this article is highly suitable with regard to the requirements for WMSNs operating in the smart grid environment.

Influence of parameters

Many parameters in the PTFQM can affect the performance of the system as a whole. Here, we analyze the influence of the queue length for level $1$ messages on channel $1$ ( $L_{1, 1}$ ) and the number of priority levels ( $n$ ).

To analyze the influence of $L_{1, 1}$ , we consider the following default parameter values:

$n = 3$ , $λ_{1} = 0.01 - 0.9$ , $λ_{2} = 0.85$ , $λ_{3} = 0.9$ , $μ_{1} = 0.95$ , $μ_{2} = 0.93$ , $μ_{3} = 0.91$ , $α_{1} = 5.85$ , $α_{2} = 2.45$ , $α_{3} = - 2.4$ , $L_{1, 1} = 3, 5, 7, 9$ $L_{2, 1} = 2$ , $L_{2, 2} = 5$ , and $L_{3, 1} = L_{3, 2} = L_{3, 3} = \infty$ .

From Figure 7(a), we can see that the overall trends of ${\bar{D}}_{1}$ for different values of $L_{1, 1}$ are essentially the same; in all cases, the delay increases with increasing $λ_{1}$ , and the increase in ${\bar{D}}_{1}$ become more gradual when $λ_{1} > 0.55$ . Importantly, however, in the curves for $L_{1, 1} = 7$ and $9$ , when $λ_{1} > 0.65$ , the average delay for level $1$ messages actually declines to some degree as $λ_{1}$ increases. The reason for this is that when $L_{1, 1}$ is sufficiently large and the arrival of level $1$ messages is sufficiently infrequent, level $1$ messages are not constantly overflowing from channel $1$ . However, when $λ_{1} > 0.65$ , level $1$ messages will preempt resources from other channels more often, and the level $1$ messages on channels $2$ and $3$ will experience less delay than those on channel $1$ , causing the overall trend of ${\bar{D}}_{1}$ to decline. In general, the ${\bar{D}}_{1}$ will increase with increasing $λ_{1}$ . And with the increase in $L_{1, 1}$ , ${\bar{D}}_{1}$ will grow faster, that is, the ${\bar{D}}_{1}$ in larger $L_{1, 1}$ condition will be greater when $λ_{1}$ is the same.

Figure 7.

Influence of the parameter $L_{1, 1}$ on ${\bar{D}}_{1}$ and $PL R_{1}$ : (a) influence of $L_{1, 1}$ on ${\bar{D}}_{1}$ and (b) influence of $L_{1, 1}$ on $PL R_{1}$ .

Figure 7(b) shows that the effect of $L_{1, 1}$ on the $PLR$ is not obvious; the $PLR$ for level $1$ messages remains at a low percentage for all considered values of $L_{1, 1}$ . Under normal circumstances, the $PLR$ will be smaller when $L_{1, 1}$ is greater, as there will be a larger space for storing packages, and fewer packets will be discarded directly.

With respect to the influence of $L_{1, 1}$ , the values of several basic parameters will vary with variations in $n$ ; although these effects are not enumerated here, Table 2 shows the average delays for all levels for values of $n$ varying from $2$ to $7$ . In this table, the shorter delays ensured for high-priority messages are indicated by the use of bold font. Clearly, the average delays for high-priority messages remain at quite low values, and degradation of the speed of service occurs only at a few low-priority levels. The variations in $n$ almost has no impact on the level 1 packages, as level 1 packages always be served priority. And as $n$ increases, the packages in level 2 to level 5 will become middle level step by step, and the average delay of level 2 to level 5 will become very small when they become middle level, this is because the package from middle level can occupy the resources from low levels. And the average delay of low level will increase when $n$ becomes bigger.

Table 2.

Average delays for various $n$ .

$n$	2	3	4	5	6	7
${\bar{D}}_{1}$	5.9018	6.0227	5.9642	5.9398	5.8946	5.8835
${\bar{D}}_{2}$	8161.3708	8.3664	8.1877	8.1591	8.0681	8.0613
${\bar{D}}_{3}$	–	20,169.7974	11.4457	9.0980	8.6569	8.6495
${\bar{D}}_{4}$	–	–	39,252.1711	2628.9300	11.2508	9.3929
${\bar{D}}_{5}$	–	–	–	64,026.4900	17,783.1789	121.9999
${\bar{D}}_{6}$	–	–	–	–	8.61E+04	3.89E+04
${\bar{D}}_{7}$	–	–	–	–	–	1.03E+05

Conclusion

In this article, we introduced the PTFQM, a new QoS-aware and hybrid-priority-based queuing model for the challenging wireless communication environment of the WMSN in smart grid. The proposed model can process multiple levels of information in accordance with their priorities, supplying superior service for messages of high priority while ensuring a certain degree of fairness for messages of low priority. To assign the appropriate priorities to the enormous amounts of data that are exchanged in the smart grid environment, we proposed an innovative prioritization mechanism, which serves as the foundation of the PTFQM. Using this prioritization mechanism, we can divide the data into subdivided QoS levels to obtain a prioritization scheme that is better adapted to the smart grid environment. The results of formula derivations and extensive simulations showed that the PTFQM outperforms the traditional NP model in providing high-priority data with superb service characterized by extremely low delays and high reliability, and in comparison with a typical DiffServ model, PTFQM can be dominant in reliability and the delays in the middle-low priorities. An implementation of the PTFQM in a MATLAB simulation showed that the performance requirements for a queuing model that is suitable for WMSN in the smart grid communication environment can easily be met and demonstrated the superior efficiency of our proposal over the traditional NP model and the typical DiffServ model. With the tuning of the value of $n$ used in the prioritization mechanism, the advantages of the PTFQM become more prominent, and its performance can be perfectly optimized for smart grid applications.

In future work, by establishing buffer lanes in various channels, we will strive to further enhance the performance for high-priority data, by the mean time ensure the performance for low-priority data, thereby enhancing the fairness of the system without excessively impacting the performance for high-priority data. The QoS support performance may also be improved by utilizing additional decision parameters for packet prioritization, such as remaining energy or buffer load. Although the PTFQM is designed for the smart grid environment, the concept of tidal flow lanes can also be adapted to other WMSN scenarios to build more suitable QoS frameworks for satisfying multi-priority performance requirements in various areas of communication.

Footnotes

Academic Editor: Miltiadis Lytras

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This paper was supported by the Director Funds of Laboratory of Network System Architecture and Convergence (2017BKLNSAC-ZJ-06).

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