in a Finite-Buﬀer Queueing Model with Generally Distributed Setup Times

. Time-dependent queueing delay (virtual waiting time) distribution conditioned by the initial level of buﬀer saturation is considered in a ﬁnite model with Poisson arrivals, generally distributed service times and setup times preceding the ﬁrst processing in each busy period. Applying theoretical approach based on the idea of embedded Markov chain, integral equations and some results from linear algebra, a compact-form representation for the Laplace transform of queueing delay distribution is obtained. Analytical results are illustrated via numerical considerations conﬁrmed by process-based discrete-event simulations.


Introduction
Queueing systems with different types of restrictions in access to the service station (server) are being intensively studied nowadays, in view of their use in modeling many phenomena occurring in technical sciences and economics. Particularly important here are models with a limited maximal number of customers (packets, calls, jobs, etc.), which naturally can describe systems with losses due to buffer overflows (buffers of input/output interfaces in TCP/IP routers, accumulating buffers in production systems). In many practical systems, which can be described by queueing models, a mechanism of turning off the server at the time when the system becomes empty is implemented; the server is being activated when the first customer arrives after the period of inactivity. The use of such a mechanism is often being forced to save energy that the server uses to remain on standby despite the lack of applications in the system (wireless networks, automated production lines, etc.). It happens quite often that the waking up of service station (restart) is not simultaneous with the start of processing in "normal" mode. The server may indeed need some time (usually random) to achieve full readiness to work. Assuming randomness of setup time, such a mechanism could be called probabilistic waking up the server. For example, a node of wireless network working under the Wi-Fi standard (IEEE 802.11) wakes thereby regularly just before sending the beacon frame from the access point [7,8].
In [6] M/G/1-type queuing system with server vacations and setup times is used to model sleeping mode in cellular network. A similar phenomenon can also be observed e.g. in production lines: after restarting, a machine needs a certain, often random, time to achieve its full readiness to work. Furthermore, the formula relating with waiting time in stationary state of GI/G/1-type queues with setup times can be found in [2,3].

Mathematical Model
In this section we state mathematical description of the considered queueing model and introduce necessary notation and definitions. So, we deal with the finite M/G/1/K−type model in which packets (calls, jobs, customers, etc.) arrive according to a Poisson process with rate λ and are processed individually, basing on the FIFO service discipline, with a CDF (=cumulative distribution function) F (·). The system capacity is bounded by a non-random value K, i.e. we have a finite buffer with K −1 places and one place reserved for service. Every time when the system becomes empty the server is being switched off (an idle period begins). Simultaneously with the arrival epoch of the packet incoming into the empty system, a server setup time begins, which is generally distributed random variable with a CDF G(·). The setup time is needed for the server to reach full ability for job processing, hence during setup times the service process is suspended. Let f (·) and g(·) be LSTs (=Laplace-Stieltjes transforms) of CDFs F (·) and G(·), respectively, i.e. for Re(s) > 0 Let us denote by X(t) the number of packets present in the system at time t (including the one being processed, if any) and by v(t) the queueing delay (virtual waiting time) at time t, i.e. the time needed for the server to process all packets present at time t or, in other words, waiting time of hypothetical (virtual) packet arriving exactly at time t. Introduce the following notation: for the transient queueing delay (tail) distribution, conditioned by the initial level of buffer saturation. We are interested in the explicit formula for the LT (=Laplace transform) of V n (t, x) in terms of "input" characteristics of the system, namely arrival rate λ, system capacity K, and transforms f (·) and g(·) of service and setup time distributions. We end this section with some additional notation which will be used throughout the paper. So, let and introduce the notation H(t) where H(·) is an arbitrary CDF. Moreover, let I{A} be the indicator of random event A.

Integral Equations for Transient Queueing Delay Distribution
In this section, by using the paradigm of embedded Markov chain and the formula of total probability we build the system of equations for conditional timedependent virtual delay distribution defined in (2). Next, we build the system for Laplace transforms corresponding to the original one. Assume, firstly, that the system is empty before the opening, so its evolution begins with idle period and the setup time begins simultaneously with the arrival epoch of the first batch of packets. We can, in fact, distinguish three mutually exclusive random events: (1) the first arrival occurs before t and the setup time also completes before t (we denote this event by E 1 (t)); (2) the first packet (call, job, customer, etc.) arrives before t but the setup time completes after t (E 2 (t)); (3) the first arrival occurs after time t (E 3 (t)).

Let us define
where t, x > 0, 0 ≤ m ≤ K and i = 1, 2, 3. So, for example, V 0 (t, x) denotes the probability that queueing delay at time t exceeds x and the first arrival occurs after t, on condition that the system is empty at the opening (at time t = 0). Obviously, we have Let us note that the following representation is true: Let us comment on (6) briefly. Indeed, the first summand on the right side describes the situation in which the buffer does not become saturated during the setup time, while the second one relates to the case in which a buffer overflow occurs during the setup time. Similarly, taking into consideration the random event E 2 , we find Finally we have, obviously, Referring to (5), we obtain from (6)-(8) (9) Now, let us take into consideration the situation in which the system is not empty primarily (at time t = 0), i.e. 1 ≤ n ≤ K. Due to the fact that successive departure moments are Markov times in the evolution of the M/G/1-type system (see e.g. [1]), then, applying the continuous version of Total Probability Law with respect to the first departure moment after t = 0, we get the following system of integral equations: where 1 ≤ n ≤ K. The interpretation of the first two summands on the right side of (10) is similar to (6)- (7). The last summand on the right side relates to the situation in which the first service completion epoch occurs after time t; in such a case, if n = K, the queueing delay at time t equals 0, since the "virtual" packet arriving at this time is lost because of the buffer overflow. Let us introduce the following notation: where Re(s) > 0 and 0 ≤ n ≤ K. By the fact that for Re(s) > 0 we have where we obtain from (9) v 0 (s, where we denote Similarly, denoting and where Re(s) > 0, we transform the equations (10) as follows: After introducing (19), we obtain from (18) the following equations: where 0 ≤ n ≤ K − 1, and the sequence ψ n (s, x) is defined as follows: Similarly, utilizing (19) in (14), we get In the next section we obtain a compact-form solution of the system (20) and (22) written in terms of "input" system characteristics and a certain functional sequence defined recursively by coefficients α i (s), i ≥ 0.

Compact Solution for Queueing Delay Transforms
In [4] (see also [5]) the following linear system of equations is investigated: where z n , n ≥ 0, is a sequence of unknowns and α n and ψ n , n ≥ 0, are known coefficients, where α 0 = 0. It was proved (see [4]) that each solution of (23) can be written in the following way: where C is a constant and terms of the sequence (R n ), n ≥ 0, can be computed in terms of α n , n ≥ 0, recursively in the following way: Observe that the system (20) has the same form as (23) but with coefficients α i and ψ i , i ≥ 0, depending on s and (s, x), respectively. Thus, the solution of (20) can be derived by using (24). The fact that the number of equations in (24) (comparing to (20)) is finite, allows for finding C = C(s, x) in the explicit form, treating the equation (22) as a boundary condition. Hence, we obtain the following formula (see (23)-(25)): where the functional sequence R n (s) , n ≥ 0, is defined by where n ≥ 1 and α i (s) is stated in (16). Taking n = 0 in (26), we obtain the following representation: and substituting n = 1, we get since κ K (s, x) = 0. From (29) we obtain where θ(s) Now the formulae (28) and (30)-(31) allows for writing terms of the functional sequence ψ n (s, x) , n ≥ 0 (see (21)), as a function of C(s, x). In order to find the representation for C(s, x), we must rewrite the formula (22), utilizing identities (21), (26), (28) and (30). We obtain Ri−j (s)ψj (s, x) where we denote and Finally, let us substitute n = K in (26) and apply the formulae (21), (28) and (30). We get where and Comparing the right sides of (32) and (35), we eliminate C(s, x) as follows: where the formulae for α i (s), κ i (s, x), R i (s), θ(s), Ψ 1 (s), χ 1 (s, x), Ψ 2 (s) and χ 2 (s, x) are given in (16)

Numerical Example
Let us take into consideration a node of the wireless sensor network with buffer of size 6 packets, with the stream of packets of average size 100 B arriving to the node according to a Poisson process with intensity 300 Kb/s. Hence, the λ = 375 packets per second arrive to the node and interarrival time between successive packets is equal to 2, 7 ms. Subsequently, assume, that packets are being transmitted with speed 400 Kb/s according to a 2-Erlang distribution with parameter μ = 1000, that gives the mean processing time 2 ms. Moreover, let us consider that the radio transmitter of the node is switched off during an idle period and needs an exponentially distributed setup time to become ready for processing. Consider cases in which the mean times are equal to 1, 10, and 100 ms, respectively. The probabilities of P{v(t) > x|X(0) = 0} for x = 0.001 and x = 0.01 are presented in Fig. 1. The figures show that the analytical results are compatible with process-based discrete-event simulations (DES). Open Access. This chapter is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/ 4.0/), which permits use, duplication, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, a link is provided to the Creative Commons license and any changes made are indicated.
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