Pre-Emptive Buffering Disciplines for Time-Critical Sensor Communications

We consider an M/G/1/K system with push-out scheme and multiple vacations. This model is particularly important in situations where it is essential to provide short waiting times to messages which are selected for service. We analyze the behavior of two types of messages: one that succeeds in transmission and the other that fails. We derive the Laplace-Stieltjes transform of the waiting time distribution for the message which is eventually served. Finally, we show some numerical results including the comparisons between the push-out and the ordinary blocking models.


Introduction
This paper considers a queueing system with a finite buffer and server vacation.Messages are admitted into the system in accordance with an appropriate buffering policy.That is, a finite number of messages can be held in the system at any time since the system contains a finite capa- city of a buffer.There are two control policies for processing messages.One is the buffering policy by which messages are selected for admission into the system.The other is the service policy by which messages are selected for admission into the service facility.
Buffering policies specify those messages that are admitted to enter and those to be removed from the buffer when the buffer is full.Rubin and Ouaily [6] classified the buffering policies into the following types (Fig. 1).
Non-Preemptive-Buffering (NPB) An arriving message that finds the system full is blocked.
Preemptive-Buffering (PB) If an arriving message finds the system full, the message which has waited the longest time is pushed out from the buffer and the arriving message is allocated a buffer space.
The service policy determines the selection of messages waiting for service when the service facility becomes available.This policy includes, for example, First-Come First-Served (FCFS), Last-Come-First-Served (LCFS), and random order of service.
Queueing systems with a finite buffer and server vacation have been extensively studied to model and analyze a number of computer communication systems.In particular, queueing sys- tems with buffering policy have many applications like time-critical message transmission, sensor telemetry, radar communication and processing systems.In those applications, the information content of a message is associated with a timeliness index, so that the most recent message to arrive contains the most valuable information, and thus needs to be given preference for selection for service.On the other hand, the data transmission is the primary job for those systems and, when there are no messages in the buffer, they start secondary jobs like testing and maintenance work.From a queueing theoretical point of view, those periods spent for the service of secondary jobs are considered as vacations.
Recently, with the increase of demands for multi-media communication, many protocols and architectures to accommodate traffics of different characteristics from multiple sources in a com- mon channel have been proposed and implemented so far [1,8].In this communication environ- ment, messages are classified from two orthogonal points of view, delay and loss probability [7].Delay (loss probability) sensitive messages are insensitive to loss probability (delay) in general.
These factors can be expressed by assigning a timeliness index to each message, which means after some critical value for its delay, each message becomes useless.For effective transmission of two types of messages, switching systems require the use of finite pre-emptive buffering service system since it is essential to provide short waiting time to those messages which are delay sensitive.If we focus our attention on the behavior of delay sensitive messages, the transmission of loss sensi- tive messages are considered as a secondary job for those switching systems.Thus, we can apply our model to evaluate the behavior of delay sensitive messages.
There are several literatures concerning buffering policies.A communication system under a pre-emptive buffering was investigated by Rubin and Ouaily in the context of an M/G/1/K with push-put scheme [6].(A technical error in the analysis of Rubin and Ouaily is pointed out in this paper.)Krhner analyzed loss probabilities for a partial buffer sharing scheme under FCFS [4].
Sumita and Ozawa analyzed loss probabilities and the waiting time of systems with a push-out scheme [7].
Concerning queueing systems with server vacation, there are a number of previous works.An excellent survey of queueing systems with vacations, including some applications, was written by Doshi [2, 3].An M/G/1/K with multiple vacation has also been analyzed by Lee [5], but not analytical results are available for the model with push-out scheme.This paper is organized as follows: In section 2, we describe our mathematical model in detail.In section 3, we derive the relation of the mean waiting times for NPB, PB-served and PB-pushed-out messages.We also summarize Lee's results [5] to obtain the joint probability dis- tributions for the number of messages in the system and the remaining service or vacation time.
In section 4, the Laplace-Stieltjes transform (LST) of the waiting time distribution for an even- tually served message is derived.In section 5, we show the numerical results.

Model
We consider an M/G/1/K push-out model with multiple vacations (Fig. 2).Messages arrive at the system according to a Poisson process with rate ,.The service time distribution function and its LST are denoted by S(x) and S*(s), respectively.The mean service time is 1/#.
When the system becomes idle, the server takes a vacation.The vacation policy of our model is multiple vacations.The server takes vacations repeatedly until it finds at least one waiting message accommodated upon returning from a vacation.The vacation titne distribution function and its LST are denoted by V(x) and V*(s), respectively.The mean vacation time is 1Iv.
The maximum number of messages that can be present in the system is K( < oc).When a message is in service, the maximum number of messages in the buffer is K-1.The buffering policy determines which to discard out of K-1 messages (K messages) to accommodate a newly arriving message when the server is busy (taking a vacation) and the system is full.
We consider the following buffering policy: When a new message finds the system full, message with the longest sojourn time in the buffer is pushed out and lost.
We deal with two service disciplines, FCFS and LCFS.
3. Queue Size Distribution and Mean Waiting Time 3.1 Queue size distribution Since the message loss happens only when the system is full, the queue length distributions in the NPB and the PB schemes are identical.Thus, we can apply Lee's results for the NPB model to the PB model [5].This section summarizes his results.
We choose a set of imbedded Markov points at those points in time when either a service is completed or a vacation ends.Let L n be the number of messages in the system immediately after the nth Markov point, and let 0 if a vacation ends, r/n 1 if a service is completed, (1) at the nth Markov point.We consider the limiting probability distributions which satisfy the following equations wk-(wo+ro)fk 0_<k_<K-1, rn gfm k + 1 l(cOj n t-7r j)ak j + 1, 7r _,j= (5) From (2) and (3), we can obtain wk(k O,...,K) and rk(k O,...,K-1).
Let L, p' and P B denote the queue size, the carried load and the probability that an arriving message is blocked in the NPB model, respectively.The carried load p' and the blocking probability P B are given by p,= where p A/#.
The joint distribution of the state of the server, the number L of messages in the system, and the remaining vacation time V when the server is on vacation or the remaining service time S when the server is busy at an arbitrary instant is given as follows.If we define the state of the server as 0 the server is on vacation, -1 the server is busy, (s)   and also define the following equations (s) we obtain (see [5] for details) For later use let k(x) and Hk(x be the inverse transforms of LST and IIk(s), respectively.

Mean waiting time
Following the approach of [6], we consider the relation between the mean waiting time of NPB model and that of PB model.Let W p denote the waiting time during which a message stays in the buffer in PB model.We then have Let 3' denote the system throughput.In both NPB and PB models, the event that a message is lost occurs when the system is full.Note that the stochastic behavior of the number of messages in the system does not depend on our buffering policy.Hence, L, 7 and p' are invariant in the NPB and PB models with multiple vacations.Let W B be the waiting time of a message accepted in the NPB model.Applying Little's theorem to those messages present in the queue, we have 7E[WB]-E[L]-p'-AE [Wp]. (15) Since 7 <_ , it follows that Considering the throughput 7, we have 7 ,(1 PB)  Substituting ( 18) and ( 19)into ( 14 (20) From ( 15)and ( 17), we obtain E [Wp] (1 PB)E[WB]. (21) From ( 20) and ( 21), E[WpIpushed-out is given by Thus, we can calculate the mean sojourn time of a pushed-out message from (22) if we obtain E [Wp served].
4. Waiting Time Distribution for Served Messages 4.1 FCFS systems We first consider the push-out system under FCFS service discipline.Each arriving message joins the queue at the tail and if the system is full upon arrival, the message at the head of the queue is pushed-out.
Let W k. n denote the waiting time of a tagged message that has k other messages ahead and n others behind it at the end of a service or a vacation.We also define the following LST, where 0_<k_<K-1 and 0_<n_<K-k-1 at the end of a vacation, and 0_<k_<K-2 and 0 _< n _< K-k-2 at the end of a service.Note that the LST Wk:n(s) is the same in both cases of a vacation and a service.

(26)
Using the above LSTs, the LST W*(s) of the distribution function for the waiting time of a served message in the FCFS system is given by a.(), w. (29) In [6], there is a technical error.The waiting time distribution of a served message W(t) is given by K W(t) 0 + E n [R(t)*B(n-1)(t)]' n=l where n'S are the steady state probabilities that an arriving message finds n messages in the system, B(t) is the service time distribution, R(t) is the remaining service time distribution, denotes the convolution and B (n-1)(t) is the n-lth-convolution.In that equation, the number of messages at an arriving epoch and the remaining service time are treated as being independent, but that is incorrect.The number of messages at an arriving epoch is not independent of the remaining service time.Thus, we have to use the joint distribution of the number of messages and the remaining service time.(We show the corrected LST of the waiting time distribution in the Appendix.) 4.2 LCFS systems We next consider the LCFS system.Each arriving message joins the queue at the head and, if the system is full, the message at the tail is pushed-out.
As in the case of FCFS, let W k denote the waiting time of a tagged message that has k other messages ahead at the end of a service or a vacation.We define the following LST, where 0 < k < K-1 at the end of a vacation, and 0 < k < K-2 at the end of a service.that the LST Wk(s is the same in the cases of both a vacation and a service. The set {Wk(s);0 <_ k _< K-1} satisfies the following equations" For simplicity, we define the following LSTs: where S(x)-Prob[S <_ x] and V(x)-Prob[V <_ x].If k messages arrive at the system during the remaining vacation or service time, the tagged message has k messages ahead at the end of this vacation or service.Thus, we have the LST of the distribution function for the waiting time of a served message in the LCFS, W*(s) by }2 ^* W*(s) 1--PB (1 ) * p' k=0 k=0

Numerical Results
In this section, we show the numerical results for the mean and the coefficient of variation (c.v.) of the waiting time using the analysis presented in section 4.
In our numerical examples, we choose the system size K equal to 5, that is, the buffer size equals 4. As for vacation times, we assume an exponential distribution with mean 1.0.The mean service time is fixed at 1.0, and the performance values are calculated by changing the arri- val rate.
First, we compare the mean waiting time under various situations.Using ( 22), ( 27) and ( 35), we calculate the mean waiting times for served and pushed-out messages.From [5], the mean waiting time for NPB model can be also calculated.Fig. 3 and Fig. 4 illustrate the mean waiting time for three types of messages: NPB, PB- served and PB-pushed-out.Furthermore, mean waiting times under the exponential service distribution are compared with those under deterministic one.
In both figures, the mean waiting times of NPB and PB-served messages tend to the value of 1 as the offered load gets small.This is because each arriving message most likely waits for the re- maining vacation time.On the other hand, the mean sojourn time of a pushed-out message is larger than those of others.This phenomenon can be explained as follows.When the arrival rate is very small, there are few messages in the system.Thus, most of arriving messages are eventual- ly served.However, if an arriving message is eventually pushed-out, its sojourn time becomes large due to light traffic.
Next, when the offered load gets large, the mean waiting times for all types of messages converge under exponential and constant service times, in particular, PB-served and PB-pushed- out messages converge to the same value.In both NPB and PB cases, a new arriving message which can be accommodated in the system finds four other messages (including the message in service) ahead when the arrival rate is very large.Hence, the mean waiting time of NPB messages converges to the value 4. In PB case, a new arriving message can enter the system.But there are many other new arriving messages behind that and the probability that the tagged message is eventually served gets small.Thus, the mean waiting times in the buffer of both messages become small.
In FCFS (Fig. 3), the mean waiting time of a PB-pushed-out message is bounded by 5, because each arriving message finds at most five messages ahead.On the other hand, in LCFS (Fig. 4), it may exceed 5.This is because there is no bound on the number of the messages which are served before the service of the tagged one.
In Fig. 4 the mean waiting time of a PB-pushed-out message under the deterministic service time distribution fluctuates remarkably when the arrival rate is small.It can be considered that under deterministic service distribution, the mean waiting time of a PB-pushed-out message is influenced by the loss probability and the waiting time of a PB-served one.
One more interesting observation is the relation of the mean waiting time between PB-served and PB-pushed-out messages under two service time distributions.Let W A:B[C denote the mean waiting time of a 'C' type message under 'A' service discipline and 'B' service distribution.
In Fig. 3, it is observed that WFcFS.Exp[Served < WFcFS:Exp[Pushed-out], i.e., the mean waiting time of a served message is always smaller than that of a pushed-out one.On the other hand, under the deterministic service time distribution, we see that (37) In the LCFS case, we can observe the following relations, Equations ( 38) and (39) show that the mean waiting time of the served message is always smaller than that of the pushed-out one under both service distributions.Thus, in FCFS, the mean wait- ing times of the served and pushed-out messages are more influenced by the type of service distri- bution.
In Fig. 5 and Fig. 6, mean waiting times are compared for two-pushed-out models: the system with vacation and that without vacation.We can calculate the mean waiting time of the system without vacation by [6] (see Appendix).In both figures, we assume S(x) to be exponential (mean service time 1.0).From both figures, we can observe the influence of vacation when the offered load is small.Furthermore, when the offered load becomes large, each mean waiting time converges to the same value.This is because taking vacations hardly affects the performance measures when the offered load is large.Fig. 7 illustrates the c.v. of the waiting time of the PB-served message under two service time disciplines and two service distributions.In both FCFS and LCFS cases, the values start from 1 because the vacation distribution is exponential and its mean equals 1.0.We also observe that both curves converge rapidly.This means that the fluctuation of the waiting time is small when the offered load becomes large.We note that the variation under LCFS is larger than that under FCFS.
Fig. 8 illustrates the c.v. of the waiting time for NPB and PB-served messages with and with- out vacation.When the offered load is small, the influence of vacation is recognized.In FCFS cases, c.v.'s converge to the same value when the offered load is large.On the other hand, in LCFS cases, c.v.'s of the PB-served message with and without vacation converge to the same value but that of NPB model diverges to infinity.We observe that the waiting time of the PB- served message with vacation varies least in both FCFS and LCFS cases.

Conclusion
In this paper, we have considered a buffer controlling policy, called push-out scheme.We investigated the behaviors of the two types of messages, one is eventually served and the other is pushed-out from the system.

Figure 6 "FCFS
Figure 3" Mean Waiting Time under FCFS