Loss rates in the single-server queue with complete rejection

Consider the single-server queue in which customers are rejected if their total sojourn time would exceed a certain level K . A basic performance measure of this system is the probability P K that a customer gets rejected in steady state. This paper presents asymptotic expansions for P K as K → ∞ . If the service time B is light-tailed and interarrival times are exponential, it is shown that the loss probability has an exponential tail. The proof of this result heavily relies on results on the two-sided exit problem for L´evy processes with no positive jumps. For heavy-tailed (subexponential) service times and generally distributed inter-arrival times, the loss probability is shown to be asymptotically equivalent to the trivial lower bound P ( B > K ).


Introduction
This paper considers the following variation of the single-server queue: customers that arrive are accepted if and only if their total sojourn time is less than a fixed constant K. If this is not the case, then a customer is rejected completely. Thus the workload W K,n in the system before the n-th arrival is driven by the following recursion: We are interested in the probability P K that a customer is rejected in steady state, more precisely, in the behavior of P K as K → ∞. If the system load ρ < 1 (which we assume throughout this paper) it is clear that P K → 0. This paper gives exact rates of convergence for both light-tailed and heavy-tailed service times. The model described by (1.1) seems to have a special place in the literature on queueing models with rejection. In particular, it is not as well understood as the single-server queue where customers are not completely but only partially rejected (i.e. part of a rejected customer's work is accepted such that the buffer is completely filled); this model is also known as the finite dam. The steady-state distribution of the workload in this queue is already known since Takács [26]. The probability P p K that a customer is (partially) rejected can be expressed in terms of the tail distribution of the maximum amount of work V max in the system during a busy cycle of the infinite buffer queue. In particular, the following result (which holds for the GI/G/1 queue with partial rejection) can be found in Bekker & Zwart [7]: (

1.2)
Another tractable model is the M/G/1 queue where customers leave the system due to impatience when their waiting time has exceeded a fixed threshold K. In this case, the probability of abandonment P i K is equal to with W M/G/1 the steady-state waiting time distribution in the M/G/1 queue, see Boots & Tijms [10]. These formulas can easily be applied to obtain asymptotic expansions for P p K or P i K , since the asymptotic behavior of P (W M/G/1 > K) and P (V max > K) is well known for both the light-tailed and the heavy-tailed case. Unfortunately, such a simple program cannot be carried out for the single-server queue with complete rejection, even when we restrict ourselves to the M/G/1 case. The main problem is the intractable distribution of the amount of work in the system when a customer is rejected. (In the case of partial rejection, this amount of work is always K.) Another problem with this queueing model is that its driving recursion (1.1) fails to be monotone in its main argument W K,n . This rules out the possibility of relating P K to a first passage probability using the framework of Asmussen & Sigman [2]. This approach has been proven quite fruitful when considering queues with partial rejection; see e.g. [7]. Nevertheless, special treatments are possible for the M/M/1 and M/D/1 queues; see Cohen [11], Gavish & Schweitzer [17] and Asmussen & Perry [1]. De Kok & Tijms [19] derived the asymptotic behavior of P K in the M/M/1 case with service rate µ. In particular, they show that For the more general M/G/1 queue, it is conjectured in [19] that P K has an exponential tail. This conjecture was only partially resolved by Van Ommeren [21], who obtained asymptotic lower and upper bounds.
The main goal of the present paper is to settle this conjecture for a general class of lighttailed service-times: it is shown that, for some constants D and γ, as K → ∞. Unfortunately, the prefactor D in this expansion is quite difficult to compute. The expression we obtain for D is related to the solution of a certain Fredholm-type integral equation. This result should be contrasted with the case where service times are heavy tailed (more precisely, when service times are in the class S * , see Section 2). In that case we show (for the GI/GI/1 queue) that Thus, the trivial lower bound P K ≥ P (B > K) is attained as K → ∞.
Not surprisingly, the methods we use to prove the asymptotic expansions for P K strongly depend on whether service times are light-tailed or heavy-tailed. In the light-tailed case, we heavily rely on results on the two-sided exit problem for completely asymmetric Lévy processes (i.e. Lévy processes with no positive or no negative jumps); this is also the main reason that we restrict to exponentially distributed inter-arrival times. In present form, these results are known since Suprun [25], who approached the problem using Wiener-Hopf factorization. The results of [25] came available to a wider audience in Bertoin [9]. The latter paper attacks the two-sided exit problem using excursion theory. A survey containing martingale proofs is Kyprianou & Palmowski [20]. More recent papers on Lévy processes, queues, and partial rejection mechanisms are [6,14,22] The results which are of direct use for us are collected in Section 4. Using these results, we are able to obtain an expression for the distribution of the amount of work right before a loss occurs. This distribution provides the key to deriving the asymptotics. When service times are heavy-tailed, the main point is to show that the system workload is O(1) (as the buffer size K → ∞) when a customer is rejected. This is possible by exploiting some estimates due to Asmussen [4] and Foss & Zachary [16]. The methods we use here allow for generally distributed inter-arrival times. This paper is organized as follows: a detailed model description of the single-server queue with complete rejection, as well as some auxiliary results on the single-server queue with infinite buffer size, are given in Section 2. We present our main results in Section 3. Section 4 is devoted to the two-sided exit problem for Lévy processes with no positive jumps. These results are then applied in Section 5 to obtain a proof of the asymptotics for P K in the light-tailed case. A proof of the heavy-tailed asymptotics can be found in Section 6. Section 7 puts our results in perspective by comparing them with other rejection disciplines.

Preliminaries
This section contains several preliminary results. We start with a description of the workload process. Then we give several asymptotic results for the single-server queue without rejection which are used in this paper.

The single-server queue with complete rejection
We develop a description of the workload process in the single-server queue with complete rejection. Let T 1 , T 2 , . . . be the inter-arrival times of the customers and denote the arrival epoch of the n-th customer after time 0 byT n , i.e.,T n = n k=1 T k . Service times are given by the i.i.d. sequence B i , i ≥ 1. A generic service time is denoted by B, and has Laplace-Stieltjes transform (LST) β(s). Throughout the paper, it is assumed that ρ = λE[B] < 1. The workload process {V K (t), t ∈ R} is then recursively defined by where I (·) is the indicator function. The workload process {V K (t), t ∈ R} is regenerative, with customer arrivals into an empty system being regeneration points. Assume that V K (0) = W K 0 = 0 and that a job arrives at time 0. Let C K = inf{t > 0 : V K (t) = 0} be the length of a busy period and N K be the number of customers served in that period. We drop the constant K from the notation if we consider the system with K = ∞. Whenever service times are light-tailed (we make this more precise later on), we also assume that customers arrive according to a Poisson process with rate λ.

The single-server queue with infinite buffer size
Our analysis partly relies on several results for the standard single server queue. In particular, we need the tail behavior of the waiting-time distribution, and the tail behavior of the distribution of the maximum workload during a busy cycle; these results are gathered in this section. As mentioned in the introduction, we consider both light-tailed and heavy-tailed asymptotics. When we assume that the service time distribution is light tailed, we mean the following: Assumption L There exists a constant γ > 0 such that If Assumption L is valid, then the tail of the waiting-time distribution in the M/G/1 queue satisfies: The constant C is given by C = (1 − ρ)/(λE[Be γB ] − 1). This result, due to Lundberg, is classical and can be found in most applied probability textbooks; see for example Theorem XIII.5.2 of Asmussen [5].
A similar result holds for the maximum amount of work during a cycle, defined as V max = max t∈[0,C] V (t). The following result is due to Iglehart [18], and is again valid under Assumption L: The above results are all concerned with light-tailed service times. In this paper we call service times heavy-tailed if they belong to the class S * , i.e.
If Assumption H holds, then the following asymptotic estimate holds for the maximum waiting time during a busy cycle W max , even for the GI/GI/1 queue; see Asmussen [4] and Foss & Zachary [16]: Foss & Zachary [16] also show a converse result: if (2.7) holds, then the service time distribution satisfies Assumption H. For background on heavy tails, we refer to the monograph Embrechts et al. [15]. Note that V max ≥ W max , and that V max ≤ W max + a if interarrival times are a.s. bounded by a finite constant a. We will use this fact in Section 6; it can in fact be used to show the following result, which we could not find explicitly recorded in the literature: follows from (2.6) and the bound V max ≥ W max . To achieve an upper bound, make the interarrival times smaller by truncating them at a finite constant a, and let N a be the resulting number of customers in a busy period; note that N ≤ N a . Since V max ≤ W max + a, since by Assumption H P (B > K) ∼ P (B > K − a), and by (2.6), we see that The proof is now completed by noting that, due to bounded convergence, as a → ∞.

Main results
In this section we present the main results of this paper, i.e. asymptotic expansions for P K under light-tailed and heavy-tailed assumptions. We first present our result for light-tailed service times. Define With these definitions we are able to state our first theorem: Theorem 3.1. Assume that the arrival process is Poisson, let ρ < 1, and assume that the service-time distribution satisfies Assumption L. Then there exists a constant D ∈ (0, ∞) such that The prefactor D can be written as with C 0 as in (2.5) and Thus, as conjectured in De Kok & Tijms [19], the probability P K indeed has an exponential tail. Unfortunately, the prefactor D is very difficult to compute; especially when using the expression given above. Recall that for the M/M/1 queue, D can be computed: can be viewed as the solution of a Fredholm-type integral equation with kernel q(x, y). The relation between such equations and queues with rejection has been observed before in [1]. A probabilistic interpretation of q(x, y) is given in Section 4.
As the next result shows, the asymptotics for P K in the heavy-tailed case are much easier to describe. Moreover, it is not necessary to consider Poisson arrivals: Assume that the arrival process is a renewal process, let ρ < 1, and assume that the service-time distribution satisfies Assumption H. Then Thus, the trivial lower bound P K ≥ P (B > K) is asymptotically exact when service times are heavy tailed. Theorem 3.2 reveals that, in the heavy-tailed case, a customer is most likely rejected since its own service time is large. Right before (thus also right after) rejection, the workload in the system is O(1) as K → ∞.
In the proof of Theorems 3.1 and 3.2 we use the following representation for P K . Let N K denote the number of customers arriving during a busy period, and let L K the number of customers lost during a busy cycle. Then, using the theory of regenerative processes, we obtain In the third equality, we used the obvious identity P (L K ≥ 1) = P (V max ≥ K).
With this representation at our disposal, the idea of the proof is clear: In both the lighttailed and the heavy-tailed case, it holds that in the M/G/1 queue). Furthermore, the asymptotic behavior of P (V max ≥ K) is given in Subsection 2.2, both under Assumption L and Assumption H. Thus, it remains to be shown that E(L K | L K ≥ 1) converges to a constant as K → ∞. In Section 6 we show that this constant converges to 1 if service-times are heavy-tailed.
Obtaining the limit of E(L K | L K ≥ 1) under light-tailed assumptions (which equals D 0 ) is much more involved. This requires several non-trivial results on Lévy processes which are given in the following section.

The two-sided exit problem
This section concentrates on the two-sided exit problem and paves the way to the proof of Theorem 3.1, which is the subject of the next section. We use the same notation as Bertoin [9]: consider a Lévy process X t , t ≥ 0, with no positive jumps. Define P x (·) as P (· | X 0 = x), and set P = P 0 . The distribution of X t is given by its moment generating function E(e sXt ) = e tψ(s) .
An important special case (in view of our queueing application) is when with (as in the previous sections) B i , i ≥ 1, an i.i.d. sequence with common LST β(s), and N t , t ≥ 0, a Poisson process with rate λ. In that case, Fix a, and define T = inf{t : X t / ∈ (0, a)}.
Let ∆ T be the jump at time T , i.e., ∆ T = X T − X T − . This section presents the joint distribution of X T − and ∆ T , both for fixed a and a → ∞. First, we treat the case of fixed a. We start with a classical result (Takács [26]): with W : [0, ∞) → [0, ∞), the unique continuous function such that The function W is known as the scale function; if X t is compound Poisson, one can relate W to the steady-state waiting-time distribution in the M/G/1 queue if the latter exists, cf. (3.1). The joint distribution of X T − and ∆ T has been given in Bertoin [9]; see also Suprun [25]. In the present paper, we only need Corollary 2 of [9], which is restated in the following proposition.
Proposition 4.1. (Bertoin [9]) For every x, y ∈ (0, a) and every z ≤ −y we have where Λ denotes the Lévy measure of X. Note that the original statement in [9] contains a typo (Λ(−y + dz) rather than the correct Λ(dz)) which is corrected here. Using this proposition, we now derive the asymptotic distribution of (X T − , ∆ T ) under the assumption that X t is of the form (4.1) and that X t has a positive drift. Under (4.1), the latter assumption is equivalent to Note that, when (4.1) holds, the Lévy measure in Proposition 4.1 is given by Using Proposition 4.1 we obtain the following result.

Proposition 4.2.
Assume that X t is compound Poisson as in (4.1) with ρ < 1 and that Assumption L holds. Then, as a → ∞, for each x,

4)
In particular, This proposition gives the asymptotic distribution of the level of X t right before jumping below 0. As one can see, the asymptotic distribution is independent of the level x, which is not very surprising.
Proof. The proof follows from direct computations. Fix x, y, z and write for a > x + y, using Proposition 4.1 and (4.2), We treat the numerator and denominator on the right hand side of this expression separately. First, we analyze the denominator. Using (2.4), it follows that, as a → ∞, (1)). (4.6) This implies, as a → ∞, To obtain the asymptotic behavior of the numerator, we apply (4.6) four times. A simple computation then gives This implies, as a → ∞, which completes the proof.
The previous result provided the asymptotic distribution when one starts at a high level a − x, i.e. close to a. We also need the asymptotic distribution as a → ∞ when we start at level x (i.e., close to 0); this is presented in the next proposition.
Recall that q(a, x, y)dy = P x (X T − ∈ dy; X T ≤ 0). We close this section with some remarks: • The function q(x, y), appearing as limit in Proposition 4.3 and already defined in Section 3, can be interpreted as follows: consider a risk process with initial capital x.
Then q(x, y)dy is the probability that ruin eventually occurs, and that the surplus before ruin is in the interval (y, y + dy). The distribution of the surplus prior to ruin has been in investigated in Schmidli [24].
• Both Proposition 4.2 and 4.3 are for compound Poisson processes. This assumption can be relaxed: asymptotics for the scale function W (x) without the assumption (4.1) can be derived from results in Bertoin & Doney [8], who prove an analogue of (2.4) for the supremum of a Lévy process. Since our primary interest is in the compound Poisson case, we omit the details.
We now turn to an analysis of the loss probability P K .

Proof of Theorem 3.1
In this section we give a proof of Theorem 3.1, which states the asymptotics for P K under Assumption L. Recall that By dominated convergence we have E[N K ] → E[N ] = 1/(1 − ρ), and from (2.5) we obtain P (V max ≥ K) ∼ C 0 e −γK . Thus, to prove Theorem 3.1, it suffices to show that, under Assumption L and ρ < 1, with D 0 defined as in Section 3. Write We now obtain an expression for P (L K ≥ n | L K ≥ 1) in terms of the density q(a, x, y), as derived in the previous section. For this, it will be convenient to work with the process R K (t) = K − V K (t) representing the spare capacity of the buffer at time t; recall that V K (t) is the workload at time t as defined in Section 2.1. Let t n be the time of the n-th rejection in a cycle. We take t n = ∞ if L K < n. Define for n ≥ 2 the following densities: Using the strong Markov property, it is obvious that for n ≥ 2, p K,n (x, y) = q(K, x, y).
Then, for n ≥ 2, It remains to specify p K,1 (y). This density is given by Finally, note that for n ≥ 2, We now let K → ∞. Using Proposition 4.3 and (5.3), we obtain We now inductively prove that the quantities p K,n (y) converge. We start with n = 1. Using Proposition 4.2 we obtain It is not difficult to show that for each y, the density of P K−u (X T − ∈ dy | X T ≤ 0) is bounded in K and x, 0 ≤ x ≤ K. Thus, using the bounded convergence theorem, we obtain p K,1 (y) → p 1 (y). (5.9) From this, we readily obtain by an inductive argument: Finally, we obtain that, for n ≥ 2, Thus, since p 1 = 1, we conclude that ∞ 0 p n (y)dy. (5.12) That this quantity equals D 0 as given by (3.3) can easily be verified by iterating (5.10). This completes the proof of Theorem 3.1.

Proof of Theorem 3.2
In this section, it is assumed that Assumption H is in force. Starting point is again the expression Since, cf. (2.7), , it suffices to show that To prove this, we first assume that inter-arrival times are bounded by a finite constant a, allowing us to use an estimate due to Foss & Zachary [16]. To avoid a notational burden, we do not keep track of a in our notation for the moment.
), see also [16] for background. Recall that t 1 is the first time a customer gets rejected. Let n 1 be the first customer that gets rejected.
Note that the finite system model and infinite system model coincide up to time t 1 . For the infinite system model, let n(K + a) be the first customer experiencing a waiting time larger than K + a. Note that P (W max > K + a) if and only if P (n(K + a) ≤ N . It is shown in [16], that In words, the customer before customer n(K + a) did not experience a long waiting time.
Since interarrival times are bounded by a, this implies Note that P (B n(K)−1 > K | B n(K)−1 > a + K − h(K)) and consequently, The convergence to 0 follows from a result in [16], while the preceeding equivalence was established in [4] Now, write We first prove that term I converges to 1 and then show that II converges to 0. In both cases it suffices to prove the upper bound, the lower bound being trivial. To achieve an upper bound, we assume that the service discipline is changed into partial rejection after time t 1 . This gives a sample-path wise increase of the workload process; thus it does not decrease the number of losses until the system empties. Denote the number of losses in the partial rejection model by L p K . It is shown in [7] that L p K | L p K ≥ 1 has a geometric distribution with rate 1/E[ . We shall use these results below.
Term I As a worst case, we take V K (t 1 ) = V K (t 1 −) = h(K). It is clear that the probablity of a loss after time t 1 and before the queue empties is o(1) as K → ∞. Given that this occurs, the number of losses after time t 1 is geometrically distributed with rate 1/E[N K ]. Thus the expected number of losses, given that a loss occurs, equals E[N K ] ≤ E[N ]. From this, we conclude that Term II Assume now, to obtain an upper bound, that the system starts at level K at time t 1 . The number of additional customers that get rejected is again geometrically distributed with rate 1/E[N K ]. Thus, as K → ∞, This concludes the proof of Theorem 3.2 if inter-arrival times are upper bounded by K. For the general case, note that truncating interarrival times in a cycle does not decrease the number of losses in a cycle, i.e. L K ≤ L a K , and apply an argument similar to that in Proposition 2.1. We omit the details.

Other rejection mechanisms
In this section we come back to the two other rejection mechanisms mentioned in the Introduction. We present asymptotic expansions for the loss probability of a customer in both cases and use these expansions to compare the various rejection disciplines. For the M/G/1 queue with partial rejection we obtain from (1.2) and (2.5) that, under Assumption L, while under Assumption H, The loss probability in the M/G/1 queue where customers leave the system impatiently when their waiting time has exceeded K is given by (1.3). Note that the total workload due to patient customers satisfies the recursion A careful analysis of this recursion may lead to asymptotic expansions for P i K as K → ∞ for the GI/G/1 queue, but this is not pursued here. Instead, we focus on the M/G/1 queue which enables us to apply the exact Formula (1.3). Combining (1.3) with (2.4) we obtain under Assumption L, that Under Assumption H, the residual service time distribution B r (having density P (B > x)/E[B]) is subexponential. This implies, using Pakes [23] or Veraverbeke [28], Combining this result with formula (1.3) then yields P i K ∼ ρP (B r > K). (7.5) From the above expressions one can make the following observations: • If we compare partial rejection and complete rejection, we observe that lim sup K→∞ P K /P p K ≤ 1 for any service-time distribution. This follows by considering the number of rejected customers during a busy cycle. This number is clearly higher under the partial rejection discipline, while the total number of customers entering the system during a busy cycle converges to 1/(1 − ρ) irrespective of the service discipline. It seems plausible that the inequality also holds for finite K.
• A comparison between P i K and P K (or P p K ) is more difficult to make. If the service times are heavy-tailed it is clear from (7.5) that P i K /P K → ∞; a similar result holds for P i K /P p K . The reason for this is that the first overshoot of level K in a busy cycle grows without bound as K → ∞, cf. a result of Asmussen & Klüppelberg [3]. Thus, for the model with impatience it follows that, when the first loss in a cycle occurs, it takes a long time to return to level K (cf. the recursion 7.3). This leads to many abandonments in a cycle implying that P i K is intrinsically larger than P (V max > K).
• If we compare the asymptotics (7.4) and (7.1) under Assumption L, the abovementioned heuristics do not apply -in fact it follows from the results in Section 4 that the first overshoot of level K in a busy cycle does converge to a proper distribution. Thus, it may happen that P p K can be asymptotically larger than P i K . In particular, it follows from (7.1) and (7.4) that lim K→∞ P p K /P i K = γ λ(1 − ρ) .
• A comparison of P K and P i K under Assumption L does not lead to simple results. For the M/M/1 queue for example, we can show that, using (1.4), P K /P i K → e −ρ /ρ, which is smaller (resp. larger) than 1 if ρ is large (small). For the M/D/1 queue we have the identity P K = P i K−D which, since P i K is decreasing in K, implies P K ≥ P i K .