Queue Size Distribution on a New ND Policy Geo/G/1 Queue and Its Computation Designs

)is paper examines a new ND policy in the discrete-time Geo/G/1 queue. Under this ND policy, the idle server restarts its service when the N and D policies are simultaneously satisfied. By two classifications of the customers, the probability-generating function and the probabilistic analysis, the steady-state queue size distributions at a departure time and an arbitrary time t are studied. Finally, the theoretical results are applied to the power-saving problem of a wireless sensor network. To improve model universality and numerical slowness, some computation designs are carried out. Under the N, D, and two ND policies, the numerical experiments are presented to obtain the optimal policy thresholds and the corresponding minimum power consumptions are compared.


Introduction
In this work, we introduce a new ND policy in the discretetime Geo/G/1 queue, obtain the steady-state queue size distribution, and conduct some computation designs in the minimum power consumption of a wireless sensor network. Under the new ND policy (called the ND policy 1), when the number of the customers in the queueing system reaches at least N and, at the same time, the sum of the service time periods of all waiting customers is greater than a given nonnegative integer D, the idle server begins to offer its service for the waiting customers.
Recently, for the ND-policy discrete-time Geo/G/1 queue, Gu et al. [1] and Lan and Tang [2] obtained the steady-state queue size distribution at an arbitrary time and its stochastic decomposition. In the above two published papers, since the service time periods of the customers who arrive during the server idle period are assumed to be independent of each other, the genuine D-policy is not actually implemented. For this flaw in [1,2], Liu et al. [3] revisited the ND policy considered in [1,2] (called the ND policy 2) in the discrete-time Geo/G/1 queue, obtained the main queueing measures, and applied the theoretical results to the power-saving optimization of a wireless sensor node. Under the ND policy 2, either of the N and D policies is first satisfied and then the service will restart. In this case, frequent setup may cause a high setup power consumption. Hence, the ND policy 2 may be ineffective when the setup power consumption is much larger than the holding power consumption of the data packet (customer) in a busy cycle. If we adopt the ND policy 1 presented by this work, which begins service when the N and D policies are satisfied at the same time, then a lower setup power consumption can be caused. e reason is that relative to the ND policy 2 in [3], the ND policy 1 can decrease the setup number of the radio server.
us, for a high setup power consumption, the ND policy 1 will consume less power than the ND policy 2 in [3]. is motivates us to study the queue size distribution of the discrete-time queue with the ND policy 1 for the powersaving optimization of a wireless sense node.
It should be noted that this work and Liu et al. [3] are two different research studies. Above all, from the definitions of two ND policies mentioned above, the ND policies considered in both research studies are different. Next, in both research studies, the expression and the concrete analysis of the same queueing measure are different. It is seen from this work and [3] that our discussion and mathematical derivation are more difficult and complex than those in [3]. Finally, in both research studies, the computation designs and the optimization comparison in the power-saving control of a sensor node are different. Numerically, the study in [3] only used the geometrically distributed transmission time to compare the superiority of the optimal N policy, D policy, and ND policy 2, while our comparison utilized arbitrary probability distributions and presented an optimization procedure for the optimal ND policy. e contributions of this paper are threefold. First, this study provides a new ND policy in the Geo/G/1 queue. Furthermore, our theoretical results agree with those in the existing N policy and genuine D policy queues (see Remark 1), which shows our work has studied the genuine D policy. Second, as far as known to the author, for the existing papers on the discrete-time queues with the N policy and genuine D policy, the queue size distribution obtained in this work is new. It is useful and important when a power consumption function is constructed to analyze the optimal ND policy at a minimum power value. ird, this research conducts some computation designs for arbitrarily distributed transmission time periods, presents a concrete optimization procedure, and numerically compares the minimum power consumption of a wireless sensor node based on the optimal N policy, D policy, ND policy 1, and ND policy 2. It is easy and practical for the network practitioners to understand and use these results.
Common methods in the queueing models include the imbedded Markov chain [4], the supplementary variable [5], the theory of matrix-geometric solution [6,7], and the probability decomposition [1,2]. ese approaches are used to analyze the queue size distribution based on the assumption of independent and identically distributed (i.i.d.) service time periods. However, in the discrete-time queues involving the D policy, the service time periods of the customers who arrive during the idle period are not independent while the service time periods of the customers who arrive during the busy period are independent (see Section 3).
is property of the service time periods makes the stochastic decomposition of the steady-state queue size [8] not hold. erefore, the above approaches become too complicated to be solved especially when they are used to deal with the queue size distribution of the D policy queues. As far as known to the author, no papers with the D-policy queues were found by means of the above methods. Lately, Gu et al. [1] and Lan and Tang [2] applied a probability decomposition method to obtain the queue size distributions in the discrete-time Geo/G/1 queues with the ND policy 2. But, they avoided the dependence of the service time periods of the customers who arrive during the idle period, and the D policy they considered was not a genuine D policy, which was pointed out by Liu et al. [3,9].
Our analysis method is based on the difference of the service time periods of the customers arriving during the idle and busy periods. By two classifications of the customers, the probability-generating function and the probabilistic analysis, the steady-state queue size distributions at a departure time and an arbitrary time t + are studied, which cannot usually be obtained by the common methods mentioned above. e results for two special cases (see Remark 1) and numerical experiments show that our analysis method is effective and available for the complex discrete-time queues involving the genuine D policy and this method operates in the continuous situation as well. e rest of this paper is organized as follows: Section 2 presents the model description and some notations. In Sections 3 and 4, based on some preliminaries, we analyze the queue size distribution. In the energy-saving problem of a wireless sensor network, Section 5 conducts some computation designs and then numerically compares the superiority of the optimal N, D, and two ND policies at a minimum power consumption. Conclusions are drawn in Section 6.

Model Description and Some Notations
e model discussed in this paper is under the LAS-DA (late arrival system with delayed access) setup [10]. at is to say, a potential arrival can only take place in (t − , t), t � 0, 1, 2, . . ., and a potential departure can only occur in (t, t + ), t � 1, 2, . . ., where t − represents the instant immediately before t and t + represents the instant immediately after t. We do not permit a departure at such a time point for a customer who has just arrived at the instant previously to an empty system. e various time epochs involved in our model can be viewed through a self-explanatory figure (see Figure 1). e detailed description of the model is given as follows: (1) e interarrival time periods of customers, denoted by τ i , i � 1, 2, . . ., are independent and geometrically distributed with probability mass function (PMF) Pr τ i � k � p(1 − p) k− 1 , k ≥ 1, and probabilitygenerating function (PGF) τ(z) � (pz/1 − (1 − p)z), |z| ≤ 1. (2) e service time periods, S n , n � 1, 2, . . ., are independent and identically distributed (i.i.d.) with PMF s(x) � Pr S n � x , x � 1, 2, . . . , (s(0) � 0) and PGF S(z) � ∞ x�1 s(x)z x , |z| ≤ 1. e mean service time E(S) and the second moment E(S 2 ) are finite. e customers are served based on their arriving order. e server can serve only one customer at a time. A customer that arrives and finds the server busy must wait in the queue until the server is available. All the customers arriving at the system are assumed to be eventually served, i.e., the traffic intensity ρ � pE(S) < 1.
(3) e idle server resumes its service if the following two conditions are simultaneously met (ND policy 1): (i) at least N customers accumulate in the system and (ii) the service time backlog (i.e., the sum of the service times) of all waiting customers exceeds a given nonnegative integer D.
When N � 1, the ND policy 1 becomes the pure D policy, and when D + 1 ≤ N, we always get the pure N policy. Hence, we discuss the case of 1 ≤ N ≤ D +

2
Discrete Dynamics in Nature and Society 1, where N � 1, 2, . . . and D � 0, 1, 2, . . . so that the N policy and D policy both control the startup of the server. We use some notations as follows: the probability mass function (PMF) of the k-fold convolution of service time with itself (s (1) S 1 + S 2 + · · · + S k− 1 ≤ D < S 1 + S 2 + · · · + S k }, k ≥ 2: the probability that the threshold value D is first exceeded by the service time of the kth customer Q N,D : the number of the customers at the start of a busy period Φ N,D : the service time backlog of all waiting customers at the start of a busy period

Preliminaries
For convenience of analysis, we define the idle period, denoted by I N,D , as the time length that starts when the system becomes empty and terminates when the idle server starts to serve customers. e busy period, denoted by B N,D , represents the time length which starts when the idle period completes and ends when the system is empty. e sum of the busy period and the next idle period is defined as the busy cycle, denoted by C N,D .
In the model description, although the service time periods in the idle and busy periods are assumed to be i.i.d., the D policy determines that the service time periods of the customers who arrive during the idle period, S 1 , S 2 , . . . , S Q N,D , are conditionally dependent. In fact, Here, two inequations show the above conditional dependence. is makes the queue size distribution not be derived by the stochastic decomposition of the queue size [8], and the analysis is difficult by the common methods.
To get the queue size distribution, we denote the customer who arrives during the idle period as IC and express the customer who arrives during the busy period as BC.
rough this classification, we will derive the PGF and mean of the queue size distribution. Above all, we study the queue size and service time backlog at the start of a busy period and the idle and busy periods.

e Queue Size at the Start of a Busy Period.
At the start of a busy period, we have k(N + 1 ≤ k ≤ D + 1) customers if and only if the D is exceeded by the kth customer and this occurs with probability S (k− 1) (D) − S (k) (D) (see the notations in Section 2). e busy period starts with N customers if and only if the sum of the service time periods of these N customers is larger than or equal to D + 1. us, the PMF and PGF of Q N,D are, respectively, given by Discrete Dynamics in Nature and Society From (2), the first and second moments of Q N,D are, respectively, obtained as

e Service Time Backlog at the Start of a Busy
Period. e service time backlog is x( ≥ D + 1) at the start of a busy period if and only if the sum of the service time periods of the N customers is equal to x, or the sum of the service time periods of the first k(N ≤ k ≤ D) customers is n, n � k, k + 1, . . . , D, and the D is exceeded by the next customer. Summing for all possible k and n yields the PMF of Φ N,D as which gives the PGF of Φ N,D as where the second term of right hand side (RHS) in (6) can be expressed as Putting (7) into (6) yields Now, using (8), we get the first and second moments of Discrete Dynamics in Nature and Society . us, we easily get the PGF and mean of the idle period as follows: where τ(z) � (pz/(1 − pz)) stands for the PGF of interarrival time periods τ i , i ≥ 1.

e Busy Period and Busy
Cycle. Define L ς as the departure point queue size of the last departing IC during the busy period (excluding this departing IC) and B ς as the remaining busy period initiating with L ς customers during the busy period. en, the busy period B N,D is written as Denote B i , i ≥ 1, as i.i.d. random variables and represent the busy period initiating with a customer in the classical Geo/G/1 queue with the PGF B(z) � S(pz + pzB(z)) and mean E(B) � (E(S)/(1 − ρ)). en, according to the Galton-Watson branching process approach [11] (an effective method to analyze the busy period when the service time periods of the customers are independent random variables), the remaining busy period B ς can be decomposed as us, from the conditional mean, We have the busy period PGF as which leads to the mean busy period Discrete Dynamics in Nature and Society Using (12) and (16), we get the mean busy cycle as From (12), (16), and (17), the server is idle with probability 1 − ρ and busy with probability ρ.

The Steady-State Queue Size at an Arbitrary Time t +
In our discrete-time ND policy Geo/G/1 queue with LAS-DA, the steady-state queue size PGF L N,D (z) at an arrival time is equal to the queue size PGF Π(z) at a departure time.
By the BASTA (Bernoulli arrivals see time averages) property, L N,D (z) is equal to the queue size PGF L N,D (z) at an arbitrary time t + . Hence, we have Let p IC and p BC represent the probabilities that an arbitrary customer is an IC and a BC, respectively; then Denote Π IC (z) (Π BC (z)) as the steady-state queue size PGF at the departure time of an arbitrary IC (BC); then, conditioning on the customer type, we get Hence, as long as we obtain the steady-state queue size PGFs Π IC (z) and Π BC (z) at the departure time, the steadystate queue size PGF L N,D (z) at an arbitrary time t + is also derived.

Steady-State Queue Size at the Departure Time of an
Arbitrary BC. Define ς as the departure time of the last IC during the busy period and L ς as the queue size just after ς.
en, L ς is the number of BCs arriving during Φ N,D . erefore, the PGF and mean of L ς are, respectively, obtained as (20) Since the queue size process during the busy period after ς can be considered as that during the busy period of the Geo/G/1 queue which starts with L ς BCs, from the decomposition property of the Geo/G/1 queue with generalized vacations (see Takagi [12]), we have is the queue size PGF of the classic Geo/G/1 queue and is the PGF of the backward recurrence time of the discrete-time renewal process generated by i.i.d. L ς 's.

Steady-State Queue Size at the Departure Time of an Arbitrary IC.
Under the steady-state condition, the queue size at the departure time of an arbitrary IC depends on whether this IC is the last customer arriving during the idle period.
Case (a). If an arbitrary IC is the last customer arriving during the idle period (with probability (1/E(Q N,D ))), then at the departure time, the queue size left behind by this IC is exactly equal to the number of BCs that arrive during Φ N,D . Hence, the queue size PGF at the departure time of this IC is Case (b). If an arbitrary IC is not the last customer arriving during the idle period, we consider the case that this IC is the kth customer arriving during the idle period and the service time backlog including itself is x, . .. In this case, at the departure time, the queue size left behind by this IC is the sum of the following two quantities: (1) e number of the BCs arriving during x. Hence, the queue size PGF at the departure time of this IC becomes where Using (22) and (23), we get the probability-generating function (PGF) of the steady-state queue size at the departure time of an arbitrary IC as where Q N− k,N− k− 1 (z), Q 1,D− x (z), and Q N− k,D− x (z) are given in (23).

Steady-State Queue Size at an Arbitrary Time t + .
After substituting (21) and (24) into (19), we get the PGF of the steady-state queue size at an arbitrary time t + as

Mean Queue
Size. It follows from (19) that the mean steady-state queue size at an arbitrary time t + is where E(L IC ) and E(L BC ) are the mean steady-state queue sizes at the departure time of the IC and the BC, respectively. From (21), (9), (10), and (3), it is easy to obtain Discrete Dynamics in Nature and Society Similarly, using (24) and (3) and noting that In (28), using 0 n�1 � 0 (see the notations in Section 2), we have (31) us, after utilizing (29)-(31) in (28), we get Also, it is easy to find that Finally, by substituting (27), (32), and (3) into (26) and using (33), we obtain the mean steady-state queue size at an arbitrary time t + as Remark 1. (two special cases). In (34), we have the following cases: (1) When D � N − 1, noting that j k�i � 0, i > j (see the notation in Section 2) and we get the mean queue size of the N policy Geo/G/1 queue as which agrees with the result in Corollary 2 of [13]. (2) When N � 1, noting that j k�i � 0, i > j (see the notation in Section 2), we get the mean queue size of the D policy Geo/G/1 queue as Discrete Dynamics in Nature and Society which is identical to the result by Liu et al. [3] (setting N � D + 1 in equation (34) of [3]).

Computation Designs in the Optimal Power Control of a WSN
Consider a wireless sensor network (WSN) [14] used in the battlefield conditions for a military purpose, which is responsible for the intelligence on enemy troop movements, damage, casualties, and so on. is WSN is comprised of a sink node and a number of sensor nodes, in which the sensor nodes behave as both data packets originator and packets router and finally send packets to a sink node for further handling. e sensor nodes are installed with some energylimited batteries, a radio server, and a sensing unit. Because these sensor nodes are employed in the secret or hostile places, it is usually impossible to recharge or replace their batteries. erefore, it is very important for the long lifetime of a WSN to achieve a good power-saving scheme. is kind of WSN can be modelled as the discrete-time threshold policy queues because its protocols are operated according to a unit of slot, the transmission time of each packet is a random variable, and the power consumption can be rationally controlled by the threshold policy queues [3,14,15]. Here, our aim is to numerically compare the optimal power-saving policy that minimizes the power consumption under the N, D, and two ND policies.
For this purpose, we denote the following: E(B N,D ) ≡ the mean time length that the radio server is busy handling data packets E(I N,D ) ≡ the mean time length that the radio server is idle E(C N,D ) ≡ the mean time length that the radio server is during a busy cycle, which is the sum of E(I N,D ) and E(B N,D ) E(L N,D ) ≡ the mean number of data packets in the steady state In addition, we consider the following power consumption elements and average power consumption functions, whose definitions are identical to those by Jiang et al.  E(I N,D ), E(B N,D ), and E(C N,D ) are given by (12), (16), and (17), respectively, and E(L N,D ) is presented by (34). Also, in Section 5, s (k) (n) denotes the probability that the sum of the transmission time periods of k data packets is equal to n and S (k) (D) is the probability that the sum of the transmission time periods of k data packets does not exceed D.
Under the ND policy 2, the average power consumption per unit time (see equation (66) in [3]) is

Rewriting for the Mean Queue Size.
In order to accelerate numerical operation speed and avoid numerical errors, we rewrite the mean queue size in (34) as the expression that does not involve the series.

(43)
Hence, it is difficult for an arbitrarily distributed transmission time to compute the values of S (k) (D)(1 ≤ k ≤ D) and s (k) (n)(1 ≤ k ≤ n). is is due to the computational difficulties in the convolution summation of the discrete-time distribution.
If the transmission/service time periods follow a common geometrical distribution, s(i) � qq i− 1 , where 0 < q < 1 and i � 1, 2, 3, . . ., then it is easy to find that For an arbitrary discrete distribution of i.i.d. transmission time periods S n , n ≥ 1, it seems difficult to compute S (k) (D)(1 ≤ k ≤ D) and s (k) (n)(1 ≤ k ≤ n). However, from the results by Alfa [16] and Latouche and Ramaswami [17], we can also find an effective method to compute the values of S (k) (D)(1 ≤ k ≤ D) and s (k) (n)(1 ≤ k ≤ n). As three lemmas, we give the following results.
Lemma 1 (see Theorem 2.1 in Alfa [16]). If the probability distribution of the discrete random variable X is Pr X � i { } � a i , i � 1, 2, . . . , n, then X follows a discrete n-dimensional phase-type distribution PH d (α, T) with parameters α � a 1 , a 2 , . . . , a n , where I n− 1 denotes an identity matrix of dimension n − 1 and 0 1×(n− 1) and 0 (n− 1)×1 are row and column vectors of ones of dimensions n − 1, respectively.
Lemma 2 (see Alfa [16]). Suppose that random variables X and Y are independent, X obeys a discrete n-dimensional phase-type distribution PH d (α, T), and Y follows a discrete m-dimensional phase-type distribution PH d (β, S); then, X + Y obeys a discrete (n + m)-dimensional phase-type distribution PH d (c, C) with parameters where α 0 denotes the probability that X locates the absorbing state at the initial time and 1 n is a column vector of ones of dimension n.

Remark 4.
Our aim is to find the phase-type distribution of the sum of i.i.d. transmission time periods S 1 , S 2 , . . . , S n with an arbitrary discrete distribution. Since it is impossible that the transmission time locates the absorbing state at the initial time, the initial probability of transmission time α 0 is taken as zero in the following numerical experiments. [17]). Suppose that the random variable Z obeys a discrete phase-type distribution PH d (α, T); then, the probability distribution of Z is
In finding the optimal policy threshold (N * , D * ) of A i (N, D), i � 1, 2, it is not easy to analytically prove that the functions A 1 (N, D) and A 2 (N, D) are convex or unimodal because they are not linear and very complicated. erefore, we will numerically determine the optimal threshold (N * , D * ) of A i (N, D), i � 1, 2. According to the optimization theory of the dynamic programme (for details, see [18]), we can develop an effective procedure to numerically get the optimal threshold (N * , D * ) of A i (N, D), i � 1, 2. In fact, for a given positive integer N, the optimal value D * (N) of the threshold D, so as to minimize A i (N, D), is given by us, a numerical procedure to find the optimal threshold (N * , D * ) of (N, D), so as to minimize A i (N, D), i � 1, 2, can be presented as follows: Step 1.

A Numerical Example.
A numerical example is presented to illustrate how the designer or practitioner of a wireless sensor network can use the above results to make decisions regarding the optimal policy (N * , D * ) that minimizes the average power consumption. We assume the following: (i) e data packets arrive according to a Bernoulli process at a rate p � 0.2 (ii) e transmission time obeys an arbitrary discretetime distribution: s(2) � 0.3, s(3) � 0.55, and s(5) � 0.15 (iii) e s � 300, e h � 3, e b � 500, and e i � 100 rough a MATLAB code based on the results and procedure in Section 5.2, the average power consumption A 1 (N, D) is shown in Figure 2 for the Geo/G/1 queue with the ND policy 1 and varying values of N and D. Under the ND policy 1, a minimum power consumption per unit time of 353.3288 is achieved at N * � 4 and D * � 9. e average power consumption A 2 (N, D) is shown in Figure 3 for the Geo/G/1 queue with the ND policy 2 and varying values of N and D. Under the ND policy 2, a minimum power consumption per unit time of 353.3232 is achieved at N * � 4 and D * � 11.

Numerical Comparison of the Optimal Power-Saving
Policies. To illustrate the power-saving effectiveness and superiority of the N, D, and two ND policies, a comparison on the minimum power consumptions is numerically conducted in the following two cases:  Tables 1 and 2.
From Tables 1 and 2, we see the following: (1) Under a same value of ρ (or (e s , e h )) and an identical policy, four transmission time distributions with the same mean of 3 but different coefficient of variations have minor differences in the minimum power consumption. (2) Under the same value of ρ (or (e s , e h )) and an identical transmission time distribution, two ND policies have less or equal power consumption by comparison with the N and D policies. us, two ND policies are superior to the N and D policies in decreasing the power consumption of a wireless sensor node. at is to say, two ND policies are more effective than the N and D policies in the power-saving problem of a WSN. (3) Under the same value of ρ(or (e s , e h )) and an identical transmission time distribution, the power consumed by the ND policy 1 is larger, less than, or equal to that consumed by the ND policy 2 and vice versa (see Tables 1 and 2). is means that the superiority of two ND policies is uncertain and dependent on the traffic intensity, the power      consumption elements e s and e h , and the transmission time distribution. erefore, in the power-saving design of a wireless sensor network, two ND policies should be adopted to control the sleep/wake-up mode of the radio server according to the concrete network parameter setting, such as the traffic intensity, the transmission time distribution, and the power consumption elements e s and e h .
It is expected that the data results in Tables 1 and 2 are useful and helpful for the network designers and practitioners in the military and other related fields.

Conclusion
is paper studies the queue size distribution for the discrete-time Geo/G/1 queue with a new ND policy and its computation designs in the power-saving optimization of a wireless sensor node. Numerically, we illustrate the optimal policy thresholds and the minimum power consumptions under the N, D, and two ND policies. e numerical comparison illustrates that two ND policies are superior over the N and D policies from the viewpoint of the minimum power consumption. e superiority of two ND policies is dependent on the concrete network parameter setting. ese results may be useful for the network practitioners in the related fields. In the future, the discrete-time queues involving the D policy and their applications will be our further works.

Data Availability
All the data used in this paper are directly included within the tables of this paper.

Conflicts of Interest
e author declares no conflicts of interest.