On the Stationary LCFS-PR Single-server Queue: A Characterization via Stochastic Intensity

We consider a stationary single-server queue with preemptive-resume 
last-come, first-served (LCFS-PR) queueing discipline. 
The LCFS-PR single-server queue has some interesting properties and 
has been studied in the literature. 
In this paper, we generalize the previous results such that the input 
process to the queue is given as a general stationary marked point 
process and derive some formulas concerning the joint distribution of 
queue length and remaining service times of respective customers in 
the system at arbitrary time instances as well as at arrival 
instances. 
The tool for derivation is the Palm-martingale calculus; that is, the 
connection between the notion of Palm probability and that of 
stochastic intensity.


Introduction
In this paper, we consider a stationary single-server queue with preemptive-resume last-come, first-served (LCFS-PR) queueing discipline. In the LCFS-PR discipline, a newly arriving customer is immediately served upon the arrival and, if there is one in service at his/her arrival epoch, this service is interrupted and is resumed just after the new customer leaves the system. Note that the service of the new customer can be also interrupted by subsequently customers' arrivals.
The LCFS-PR single-server queue is known to have some interesting properties and has been studied in the queueing literature: When the arrival process is given by the superposition of independent Poisson processes, the LCFS-PR queue is characterized as a symmetric queue and the stationary joint distribution of the queue length and the remaining service times of respective customers in the system is derived by Kelly [8]. Fakinos [6] extends the results to the case with GI/GI inputs and derives the stationary joint distribution of the queue length and the remaining service times at arrival instances. The corresponding distributions at arbitrary time instances are then derived by Yamazaki [16], which is later extended by Fakinos [7] to the case with the service time distributions depending on the queue length. Furthermore, Takine [15] considers the LCFS-PR single-server queue fed by the multiple arrival streams governed by a Markov chain and derives the matrix-product form solution for the stationary joint distribution of the queue length and the remaining service times at arbitrary time instances.
In the current paper, we generalize the previous works such that the input process to the queueing system is described as a general stationary marked point process, and derive some formulas concerning the joint distributions of the queue length and the remaining service times at arbitrary time instances as well as at arrival instances. With the formulas obtained here, we can explain the previous results in the literature. The tool for derivation is the Palm-martingale calculus (see, e.g., Baccelli and Brémaud [1] and Brémaud [4]), that is, the connection between the notion of Palm probability and that of stochastic intensity. Recently, the author derives a general formula for the stationary workload distribution of a work-conserving single-server queue by Palm-martingale approach ( [11]), and the current work is an extension of it. This paper is organized as follows: In the next section, the input process to the queueing system is described as a stationary marked point process. The stochastic intensity kernel associated with the marked point process is then provided. In Section 3, some general formulas for the stationary joint distributions of the queue length and the remaining service times are derived. The relations with the 1 previous results are also discussed. Section 4 focuses on the distributions of the queue length and Section 5 provides the results for the stationary distributions of the workload in the system.

Description of the Arrival Process
The arrival process to the queueing system is defined on a probability space (Ω, F, P). On (Ω, F), a family of measurable shift operators {θ t } t∈R is defined and satisfies P • θ −1 t = P for t ∈ R, that is, {θ t } t∈R is stationary in P. Let N denote a point process on (R, B(R)) counting the number of time epochs at which customers arrive at and enter the system, and let {T n } n∈Z be the corresponding point sequence satisfying that is, N is P-a.s. simple and locally finite. Each arriving customer belongs to any one of classes 1, . . . , K and let C n (∈ {1, . . . , K}), n ∈ Z, denote the class of the customer who arrives at T n . Also, let S n (≥ 0), n ∈ Z, denote the service time required by the customer who arrives at T n . We assume that Due to the stationarity of {θ t } t∈R , {(T n , C n , S n )} n∈Z then forms a stationary marked point process on the real line with mark space {1, . . . , K} × R + . We further assume that the intensity of N is positive and finite, that is, λ = E[N ((0, 1])] ∈ (0, ∞). Then, the Palm probability with respect to (N, P, θ t ) is defined by where 1 A is the indicator of event A (see, e.g., [1]). The expectation with respect to P 0 N is denoted by E 0 N . Note that P 0 N (T 0 = 0) = 1 and E 0 N [T n+1 − T n ] = 1/λ for all n ∈ Z. The traffic intensity of the queueing system is given by ρ = λ E 0 N [S 0 ]. Let N k , k = 1, . . . , K, denote the sub-point process of N counting the number of arriving customers in class k, that is, N k is defined by Clearly, N = K k=1 N k , and by the definition of P 0 N in (1), the intensity of N k is given by We assume that P 0 N (C 0 = k) > 0 for all k = 1, . . . , K. Then, the Palm probability P 0 N k with respect to N k is well defined for k = 1, . . . , K, and they satisfy λ P 0 . Now, for discussions in the following sections, we define the stochastic intensity kernel associated with the marked point process {(T n , C n , S n )} n∈Z . Let {F t } t∈R denote a history to which {(T n , C n , S n )} n∈Z is adapted, that is, F t , t ∈ R, is a sub-σ-field of F such that F s ⊂ F t whenever s ≤ t and We assume that the point process N admits the F t -stochastic intensity {λ(t)} t∈R , which is P-a.s. locally integrable and satisfies where we can always assume that {λ(t)} t∈R is F t -predictable (see [3]). We also assume that {F t } t∈R is compatible with {θ t } t∈R in the sense of θ t F s = F s−t , and assume that the following conditional distributions with respect to F 0− = t<0 F t exist for k = 1, . . . , K; These conditional distributions enable us to consider the case where the distribution of the class of an arriving customer depends on the history until his/her arrival and the distribution of the service time also depends on both the history and the class. Then, F t -stochastic intensity kernel associated with For simplicity of the notation, we also write

Remaining Service Time Distributions
In the following, we assume that the stability condition ρ < 1 holds, and under this assumption, we also assume that the queueing system is in the steady state (see [9] or [1, Chap. 2]). Let L(t) denote the number of customers in the system at time t. Note that, under the stability condition ρ < 1, we have , denote respectively the class and the remaining service time of the jth oldest customer in the system at time t. Note that, since we consider the LCFS-PR queueing discipline, the oldest customer in the system is one that starts the current busy period, and the L(t)th oldest customer is the newest arriving customer in the system and in service at time t.
Indeed, A(l, k, x) is concerning with (k 1 , . . . , k l ) and (x 1 , . . . , x l ), and G j , j = 1, 2, . . ., is the mapping on {1, . . . , K} j × R j . Note that (4) represents the distribution of the system state under the time-stationary probability P, that is, the distribution observed at arbitrary time epochs. While, (5) represents the distribution of the system state observed just before arrival instances.
Proof: The proof follows a similar line to that of Theorem 1 in [11], where the stationary workload distribution in a work-conserving single-server queue was derived. A subgoal is to have the relation between P(A(l, k, x)) and P(A(l − 1, k, x)). Let N l−1,k,x denote the point process counting the number of arrivals just before which the system is in A(l − 1, k, x), that is, The corresponding point sequence is denoted by {T l−1,k,x n } n∈Z and satisfies · · · < T l−1,k,x , k, x)). Since the queueing process is stationary and λ 1 , . . . , λ K are all positive and finite, λ l−1,k,x is also positive and finite. Let P 0 l−1,k,x denote the Palm probability with respect to N l−1,k,x and let E 0 l−1,k,x be the corresponding expectation. Applying the Palm inversion formula (see [1]) to P(A(l, k, x)), we have Here, since the customers older than the customer arriving at T 0 (called customer 0) are never served until the customer 0's service is completed, we have for t ∈ [0, customer 0 is from class k l and is in service at time t with the remaining service time not greater than x l , P 0 l−1,k,x -a.s. Namely, where a ∧ b = min(a, b). Thus, noting that P 0 l−1,k,x (·) = P 0 N (· | A(l − 1, k, x)), Now, we transform the last expression in terms of the Palm probability P 0 N into an expression in terms of the time-stationary probability P via the stochastic intensity kernel. Using the conditional distributions (2) and (3) with respect to F 0− , Since 1 A(l−1,k,x) is also F 0− -measurable, applying Papangelou's formula (see, e.g., [1,4,13]) into (7), we have P(A(l, k, x)) = E λ(0) q k l Hence, noting that P(A(0, k, x)) = P(L(0) = 0) = 1 − ρ, we have (4) inductively. Equality (5) is immediately obtained from (4) by Papangelou's formula λ P 0 N (A(l, k, x)) = E[λ(0) 1 A(l,k,x) ].
When we ignore the class of each customer, Theorem 3.1 reduces to the following: where Papangelou's formula is used in the first equality and the second equality follows from P 0 N k (·) = P 0 N (· | C 0 = k). Both are used in the third equality. In addition, if E[λ(0) | L(0) = 1] = E[λ(0) | L(0) = 2] = · · · = E[λ(0) | L(0) > 0], that is the case where N is a renewal point process independent of {(C n , S n )} n∈Z , Theorem 3.1 reduces to the multiclass version of the results by Fakinos [7].

Remark 3.2:
Furthermore, if and only if the stochastic intensity kernel is independent of the system state, which is known as the lack of bias assumption for ASTA (arrivals see time averages) property (see e.g., [10]), E[λ(0) | A(l, k, x)] = E[λ(0)] = λ in (5) and the integrand of G j in (6) reduces to where F k (y) = P 0 N k (S 0 ≤ y) for k = 1, . . . , K. Then, both (4) and (5) also reduce to the well known formula for the LCFS-PR queue with Poisson arrivals (see [8]);

Queue Length Distributions
In this section, we focus on the queue length in the LCFS-PR queue and provide the formulas for various distributions. First, letting x j , j = 1, 2, . . ., go to infinity in (4) and (5), we have the following: where 0 j=1 · = 1 conventionally and Proof: It suffices to show that, when each x i , i = 1, . . . , j, goes to infinity, G j (k, x) in (6) converges to the right-hand side of (10). Noting that for any event A ∈ F and any k = 1, . . . , K, lim x1,...,xj→∞ where we use Papangelou's formula in the second equality and use P 0 , the proof is completed.
Next, we consider the joint distribution of the number of customers in each class. Let L(t) = (L 1 (t), . . . , L K (t)), where L k (t) (∈ Z + ), k = 1, . . . , K, denotes the number of class k customers in the system at time t. Then, we have the following: where e k , k = 1, . . . , K, denotes the unit vector on R K such that only the kth element is equal to one and others are zero, and Furthermore, the distribution of L(0−) under P 0 N satisfies the recursion such that Proof: Let B(l, k) = {L(0−) = l, I |l| (0−) = k}, where |l| = K k=1 l k , that is, B(l, k) represents the event such that the queue length vector is given by l and the class of the customer in service is k at time 0−. When |l| > 0, similar to the proof of Theorem 3.1, we have Similar to the proof of Theorem 4.1, we have . Thus, summing up over k = 1, . . . , K, we have (11). Also, (12) follows from Papangelou's for- Finally, when we ignore the class of each customer, both Theorems 4.1 and 4.2 reduce to the stationary distribution of the total number of customers in the system: where 0 j=1 · = 1 conventionally, and for j = 0, 1, 2, . . ., The right-hand side represents the stationary residual life time distribution of the service time of a class k customer provided that he/she arrives at the empty system. This is indeed the observation discussed in [7,14].

Remark 4.2:
We can see from (14) that, for l = 0, 1, 2, . . ., The last expression above represents the expected number of the interruptions of an (l + 1)st oldest customer's service. Indeed, an (l + 1)st oldest customer sees l customers in the system at his/her arrival epoch and the total time that the queue length is just equal to l + 1 during the sojourn time of the (l + 1)st oldest customer is given by his/her service time. Also from (14), where we use P(L(0) > 0) = ρ = λ E 0 N [S 0 ]. Now, consider the case where the service times are i.i.d. and independent of the arrival point process N , and furthermore, suppose that E[λ(0) | L(0) = 1] = E[λ(0) | L(0) = 2] = · · · = E[λ(0) | L(0) > 0], that is the case of the GI/GI/1 queue. Then, (15) reduces to and combining with (16), we can verify that the distribution of L(0−) under P 0 N reduces to the geometric distribution with parameter γ, as is derived in [6].

Remark 4.3:
Finally, if and only if the stochastic intensity kernel is independent of the queue length and the class of each customer in the system, both (8) and (9) reduce to (1 − ρ) l j=1 ρ kj , which is also the well-known result for the LCFS-PR queue with Poisson arrivals (see [8]).
When ignoring the class of each customer, Theorem 5.1 reduces to the stationary distribution of the total amount of work in a work-conserving single-server queueing system, which is derived in [11]: Remark 5.1: Corollary 5.2 is an extension of well-known Beneš's formula for the M/GI/1 queue (see [2,5]) and its extension to the GI/GI/1 queue (see [12]).