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The combined gated-exhaustive vacation system in discrete time
Performance Evaluation 49 (2002) 227–239
The combined gated-exhaustive vacation system in discrete time Dieter Fiems∗ , Stijn De Vuyst1 , Herwig Bruneel2 SMACS Research Group, Vakgroep TELIN (TW07), Ghent University, St-Pietersnieuwstraat 41, B-9000 Gent, Belgium
Abstract We consider a discrete-time gated vacation system. The available buffer space is divided into two subsequent queues separated by a gate and new customers arrive either before or after this gate. Whenever all customers after the gate are served, the server takes a vacation. After each vacation, the gate opens which causes all waiting customers to move to the buffer space after the gate. The model under investigation allows to capture performance of a.o. the exhaustive and the gated queueing systems with multiple or single vacations. Using a probability generating functions approach, we obtain expressions for performance measures such as moments of system contents at various epochs in equilibrium and of customer delay. We conclude with a numerical example. © 2002 Elsevier Science B.V. All rights reserved. Keywords: Discrete-time queue; Vacation; Interruption
1. Introduction Queueing systems with server vacations [1,2] have proven to be a useful abstraction of systems where several classes of customers share a common resource such as priority systems [3] and polling systems [4], or of systems where this resource is unreliable such as maintenance models [5], ARQ systems [6], etc. Typically, one distinguishes following classes of vacation systems [2]: For exhaustive systems, the server only takes a vacation whenever there are no more customers in the system. The gated system keeps on serving until there are no customers left that arrived before the end of the last vacation. Number limited systems pose a maximum number of customers that can be served between server vacations whereas time-limited systems pose a maximal time between vacations, either in a preemptive or non-preemptive way [7]. For both number- and time-limited systems, the server also takes a vacation whenever the queue empties before the respective maxima expire. Additionally, one distinguishes multiple vacation systems—the server takes another vacation if there are no customers in the system when it returns from a vacation—and single vacation systems—the server waits for the first arrival when there are no customers in the system upon returning from a vacation. ∗ Corresponding author. Tel.: +32-9-264-8902; fax: +32-9-264-4295. E-mail addresses: [email protected] (D. Fiems), [email protected] (S. De Vuyst), [email protected] (H. Bruneel). 1 Tel.: +32-9-264-8902; fax: +32-9-264-4295. 2 Tel.: +32-9-264-3414; fax: +32-9-264-4295.
0166-5316/02/$ – see front matter © 2002 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 6 - 5 3 1 6 ( 0 2 ) 0 0 1 1 1 - 6
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The present contribution uses the method of the supplementary variable to investigate a model combining all features of both the gated and the exhaustive vacation system. The system takes multiple vacations and the first of these has an alternative distribution [8]. The model under investigation therefore allows to capture performance for both the single-and the multiple-vacation cases. The rest of the paper is organized as follows. Section 2 gives a detailed description of our combined gated-exhaustive vacation system. The performance analysis is presented and numerically illustrated in Sections 3 and 4, respectively, whereas conclusions are drawn in Section 5. 2. Mathematical model We consider a discrete-time queueing system, i.e., time is divided into fixed length intervals or slots. The system consists of a single server and two infinite capacity queues separated by a gate as depicted in Fig. 1. During the consecutive slots, customers arrive either in the primary or in the secondary queues. Customers in both queues are served according to a first-in first-out (FIFO) service discipline. The customers in the secondary queue move to the primary queue whenever the gate opens (see further). The numbers of customers arriving in the primary and secondary queues during the consecutive slots are modeled by means of an independent and identically distributed (i.i.d.) series of jointly dependent random variables with common probability mass function a(m, n) (m, n ≥ 0) and corresponding probability generating function A(z1 , z2 ). For ease of notation, we also introduce the generating function A(z) = A(z, z) of the total number of arrivals per slot and the marginal probability generating functions A1 (z) = A(z, 1) and A2 (z) = A(1, z) corresponding to the number of arrivals per slot in the primary and secondary queues, respectively. Service of customers is synchronized with respect to slot boundaries, which implies that service of a customer cannot start before the slot following its arrival slot. Service typically takes a number of slots and the service times (in slots) of the consecutive customers are modeled by means of an i.i.d. series of positive random variables with common probability mass function s(n) (n > 0) and with corresponding probability generating function S(z). Furthermore, we shall refer to an uninterrupted sequence of service times as a busy period. From these definitions, it is clear that the mean number of arrivals per slot and the average amount of work (in terms of slots) per arriving customer is given by A (1) and S (1), respectively, due to the moment-generating property of probability generating functions. Hence, the overall workload ρ is equal to A (1)S (1) and since there is only one server, the stability condition for this system is given by ρ A (1)S (1) < 1.
(1)
Whenever there are no more primary customers in the system at the beginning of a slot, the server takes a vacation, meaning that for an integer number of contiguous slots no customers will be served. Upon
Fig. 1. Gated-exhaustive queueing system.
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returning from this vacation the gate opens, i.e., all customers in the secondary queue move to the primary at the end of the last vacation slot. If there are no customers in the system upon returning from the vacation, the server immediately takes a new vacation. The lengths (in slots) of the consecutive server vacations are modeled as a series of independent random variables with common probability generating functions V1 (z) or V2 (z) depending on whether the vacation is not or is immediately preceded by another vacation. As with the service times, we refer to an uninterrupted sequence of vacations as a vacation period. As such, the system continuously alternates between busy periods and vacation periods. Clearly, this model encapsulates the gated—only arrivals in secondary queue—and exhaustive—only arrivals in the primary queue—vacation systems with multiple (V1 (z) = V2 (z)) and single (V2 (z) = z) vacations. Further, the model under consideration also relates to priority systems as customers arriving in the primary queue receive priority over customers arriving in the secondary queue. However, if there are no more customers in the primary queue, waiting (low priority) customers in the secondary queue can jump to the (high priority) primary queue.
3. Queueing analysis 3.1. System state at random slot boundaries Let U1,k and U2,k , respectively, denote the number of customers in the primary and secondary queues at the beginning of the kth slot. Further, let Rk denote the remaining number of vacation slots following slot k if slot k is a vacation slot and let Hk denote the number of remaining service slots of the customer in service following slot k if this is not the case. Also, for ease of analysis, we introduce an additional random variable Tk , indicating whether the server is on vacation (Tk = V ) or busy (Tk = B) during slot k. The system under consideration then yields the following set of system equations: • Given Tk = V and Rk > 0, then, during slot (k + 1), the server continues its vacation: Tk+1 = V ,
Rk+1 = Rk − 1,
U1,k+1 = U1,k + A1,k ,
U2,k+1 = U2,k + A2,k .
(2)
• Given Tk = V , Rk = 0 and U1,k + U2,k + A1,k + A2,k = 0, then, slot k is the last slot of a vacation after which the system is found to be empty. As such, the server immediately takes a new vacation: Tk+1 = V ,
Rk+1 = V2 − 1,
U1,k+1 = 0,
U2,k+1 = 0.
(3)
• Given Tk = V , Rk = 0 and U1,k + U2,k + A1,k + A2,k > 0, then, the server returns from a vacation and finds the system non-empty. The gate opens and a new customer starts service: Tk+1 = B,
Hk+1 = S − 1,
U1,k+1 = U1,k + U2,k + A1,k + A2,k ,
U2,k+1 = 0.
(4)
• Given Tk = B and Hk > 0, then, the customer in service continues service in slot (k + 1): Tk+1 = B,
Hk+1 = Hk − 1,
U1,k+1 = U1,k + A1,k ,
U2,k+1 = U2,k + A2,k .
(5)
• Given Tk = B, Hk = 0 and U1,k + A1,k > 1, then, the customer in service leaves the system and a new customer starts service: Tk+1 = B,
Hk+1 = S − 1,
U1,k+1 = U1,k + A1,k − 1,
U2,k+1 = U2,k + A2,k .
(6)
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• Given Tk = B, Hk = 0, U1,k = 1 and A1,k = 0, then, the current busy period terminates as there are no customers left in the primary queue at the beginning of slot (k + 1). The server therefore starts a vacation: Tk+1 = V ,
Rk+1 = V1 − 1,
U1,k+1 = 0,
U2,k+1 = U2,k + A2,k .
(7)
Here A1,k and A2,k denote the number of arrivals during slot k in the primary and secondary queues, respectively, S denotes the service time of a random customer, and V1 and V2 denote the lengths of server vacations not immediately or immediately preceded by another vacation. It can be seen from the above system equations that the vector {U1,k , U2,k , Rk or Hk , Tk } forms a four-dimensional Markov chain, where the variable Rk only has meaning if Tk = V and Hk only if Tk = B. As such, this vector provides a complete description for the state of the system at any given slot k. In what follows, we show how to obtain the equilibrium distribution of the system state, both during busy slots and vacation slots. Now, let Nk (x, z1 , z2 ) and Mk (x, z1 , z2 ) denote the following (partial) joint probability generating functions U
−1 U2,k z2 |Tk = B] Pr[Tk = B], Rk U1,k U2,k E[x z1 z2 |Tk = V ] Pr[Tk = V ].
Nk (x, z1 , z2 ) = E[x Hk z1 1,k Mk (x, z1 , z2 ) =
(8)
The preceding system equations (2)–(7) can be used to relate these functions for slot (k + 1) to those for slot k. After some standard z-transform manipulations, we find Nk+1 (x, z1 , z2 ) =
S(x) (Mk (0, z1 , z1 )A(z1 ) − Mk (0, 0, 0)A(0)) xz1 S(x) + (Nk (0, z1 , z2 )A(z1 , z2 ) − Nk (0, 0, z2 )A(0, z2 )) xz1 A(z1 , z2 ) + (Nk (x, z1 , z2 ) − Nk (0, z1 , z2 )), x
(9)
and A(z1 , z2 ) V2 (x) (Mk (x, z1 , z2 ) − Mk (0, z1 , z2 )) + A(0)Mk (0, 0, 0) x x V1 (x) + A(0, z2 )Nk (0, 0, z2 ). (10) x Under the assumption (1) that the system reaches equilibrium, let N(x, z1 , z2 ) and M(x, z1 , z2 ) denote the equilibrium probability generating functions, i.e., N(x, z1 , z2 ) = limk→∞ Nk (x, z1 , z2 ) and M(x, z1 , z2 ) = limk→∞ Mk (x, z1 , z2 ). Letting k → ∞ in (9) and (10), and solving for N(x, z1 , z2 ) and M(x, z1 , z2 ), respectively, then yields Mk+1 (x, z1 , z2 ) =
S(x)M(0, z1 , z1 )A(z1 ) − S(x)A(0, z2 )N(0, 0, z2 ) −A(z1 , z2 )(z1 − S(x))N(0, z1 , z2 ) − S(x)M(0, 0, 0)A(0) N(x, z1 , z2 ) = (x − A(z1 , z2 ))z1
(11)
and M(x, z1 , z2 ) =
A(0)V2 (x)M(0, 0, 0) − A(z1 , z2 )M(0, z1 , z2 ) + A(0, z2 )V1 (x)N(0, 0, z2 ) . x − A(z1 , z2 )
(12)
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Clearly, as both generating functions are analytic within the unit disk, the numerator on the right-hand side of both equations (11) and (12) vanishes for (A(z1 , z2 ), z1 , z2 ), implying N(0, z1 , z2 ) = S(A(z1 , z2 ))
A(z1 )M(0, z1 , z1 ) − A(0)M(0, 0, 0) − A(0, z2 )N(0, 0, z2 ) A(z1 , z2 )(z1 − S(A(z1 , z2 )))
(13)
and M(0, z1 , z2 ) =
A(0)V2 (A(z1 , z2 ))M(0, 0, 0) + A(0, z2 )V1 (A(z1 , z2 ))N(0, 0, z2 ) . A(z1 , z2 )
(14)
From Eqs. (11)–(14) it is clear that M(x, z1 , z2 ) and N(x, z1 , z2 ) are completely determined once one determines M(0, 0, 0) and N(0, 0, z). The quantity M(0, 0, 0) can be directly obtained from the normalization condition N(1, 1, 1) + M(1, 1, 1) = 1. Using de l’Hˆopital’s rule, this yields M(0, 0, 0) =
1 − S (1)A (1) − V1 (1)A1 (0)N(0, 0, 1) . A(0)V2 (1)
(15)
One may also obtain the former expression as follows. The fraction of busy slots is given by the load S (1)A (1). As the system is either busy or on vacation, the fraction of vacation slots is given by 1 − S (1)A (1). On the other hand, M(0, 0, 0)A(0) and N(0, 0, 1)A1 (0) represent the probabilities that a new vacation starts at the end of a random vacation and busy slot, respectively. The fraction of vacation slots is therefore also given by N(0, 0, 1)A1 (0)V1 (1) + M(0, 0, 0)A(0)V2 (1). Comparison of these fractions then yields the former expression for M(0, 0, 0). Finally, to characterize the last unknown function N(0, 0, z), we can proceed as follows. First notice that for each |z2 | ≤ 1, Rouché’s theorem assures the existence of a unique value χ (z2 ) such that, |χ (z2 )| ≤ 1 and χ(z2 ) = S(A(χ(z2 ), z2 )).
(16)
As the partial probability generating function N(0, z1 , z2 ) is analytic for |z1 |, |z2 | ≤ 1, the numerator on the right-hand side of (13) vanishes for all values (0, χ(z2 ), z2 ), |z2 | ≤ 1. As an alternative to N(0, 0, z), let Q(z) denote Q(z)
A(0, z)N(0, 0, z) , A1 (0)N(0, 0, 1)
(17)
i.e., Q(z) is the probability generating function of the number of customers in the system at the end of a busy period. Eqs. (13) and (14) then yield the following functional equation for Q(z): Q(χ(z)) =
Q(z) + K(1 − V2 (A(χ(z))) V1 (A(χ(z)))
(18)
with K
A(0)M(0, 0, 0) . A1 (0)N(0, 0, 1)
(19)
Now, consider the series, x0 = 0, xi = χ (xi−1 ) for i > 0. This series clearly converges to 1 as long as χ (1) < 1 which is implied by S (1)A (1) < 1. The latter condition is always met if the system reaches equilibrium due to (1). The convergence is brought about by the fact that χ (z) proves to be a probability
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generating function. In particular, it is the probability generating function of the number of secondary customers that arrive during the sub-busy period of a primary customer (see e.g., [9]). The sub-busy period of a primary customer is defined to start from the moment this customer enters service, and ends on the first occasion in which the (primary) queue contains one customer less than at the beginning of the sub-busy period. The concept of sub-busy periods will be further exploited in the next paragraphs when studying delay times. As the series {xi } converges to 1, the series yi = K/Q(xi ) converges to K since Q(1) = 1. From Eqs. (14), (17) and (19) one easily obtains M(0, 0, 0) K V1 (A(0)) y0 = = , (20) Q(0) N(0, 0, 0) 1 − V2 (A(0)) whereas, Eq. (18) yields V1 (A(xi+1 ))yi yi+1 = . 1 + (1 − V2 (A(xi+1 )))yi
(21)
These relations then allow to calculate K numerically up to any desired precision. Note that—as in [8]— by iteration of (18), one may also obtain an explicit expression for Q(z). The unknown value K then follows from the normalization condition Q(1) = 1. Summarizing, we get Q(z2 )(V1 (x) − V1 (A(z1 , z2 )) + K(V2 (x) − V2 (A(z1 , z2 )))) M(x, z1 , z2 ) = c (22) x − A(z1 , z2 ) and N(x, z1 , z2 ) = c
S(x) − S(A(z1 , z2 )) V1 (A(z1 ))Q(z1 ) − Q(z2 ) − K(1 − V2 (A(z1 ))) z1 − S(A(z1 , z2 )) x − A(z1 , z2 )
(23)
with c=
1 − A (1)S (1) , V1 (1) + KV2 (1)
(24)
where Q(z) is implicitly defined by (18) and where one can determine K numerically using Eqs. (20) and (21). 3.2. Performance measures: queue contents and customer delay We now show how the partial probability generating functions M(x, z1 , z2 ) and N(x, z1 , z2 ) allow to evaluate the performance of the system. Let U (z1 , z2 ) denote the joint probability generating function of the number of customers in the primary and secondary queues at random slot boundaries, then, using the definitions (8), we get U (z1 , z2 ) = M(1, z1 , z2 ) + z1 N(1, z1 , z2 ).
(25)
Analogously, the probability generating function V (z1 , z2 ) of the number of customers in primary and secondary queues at departure epochs—the beginning of a slot following a slot where a customer leaves the system—is given by N(0, z1 , z2 ) V (z1 , z2 ) = (26) A(z1 , z2 ). N(0, 1, 1)
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Fig. 2. Delay experienced by an arbitrary but tagged customer arriving in the primary queue either (a) during a vacation slot or (b) a busy slot. Customers that are served during the delay time of the tagged customer are shaded.
Using the moment-generating property of probability generating functions, one easily obtains performance measures such as mean and variance of the number of customers in primary and secondary queues and such as the correlation between the number of customers in primary and secondary queues. Further let D1 and D2 denote the delay—the number of slots between the end of a customer’s arrival slot and the end of its departure slot—of a random customer that arrives in the primary and secondary queues, respectively, and let D1 (z) and D2 (z) denote the corresponding probability generating functions. Also, let U1 and U2 denote the primary and secondary queue contents at the beginning of a random customer’s arrival slot and let H (R) denote the remaining service (vacation) time given that the server is busy (on vacation) during the random customer’s arrival slot. We first consider a random “but tagged” customer that arrives in the primary queue. Such a customer receives service when all customers that arrived in the primary queue before the tagged customer are served as depicted in Fig. 2. Given that the customer under consideration arrives during a vacation slot, its delay equals the remaining vacation time, augmented by the sum of the service times of the customers present in the primary queue upon arrival of the tagged customer and by its own service time, i.e., U1 +A∗1
D1 = R + S +
Sj ,
(27)
j =1
where A∗1 denotes the number of arrivals in the primary queue during the tagged customer’s arrival slot prior to this customer. Further, Sj denotes the service time of the j th customer starting service after the tagged customer’s arrival slot and S denotes the tagged customer’s own service time. Similarly, given that the customer arrives during a busy slot, its delay equals the remaining service time of the customer in service upon arrival of the tagged customer, augmented by the sum of the service times of the customers present in the primary queue upon arrival of the tagged customer and by its own service time, i.e., U1 +A∗1 −1
D1 = H + S +
j =1
Sj .
(28)
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Fig. 3. Delay for a tagged customer arriving in the secondary queue during a vacation slot.
Due to the independent nature of the arrival process, the state of the system at the beginning of a random customer’s arrival slot has the same stochastical properties as the state of the system during a random slot and therefore, D1 (z) = (M(z, S(z), 1) + N(z, S(z), 1))A∗1 (S(z))S(z),
(29)
where A∗1 (z) denotes the probability generating function corresponding to A∗1 (see, e.g., [10]), A∗1 (z) =
A1 (z) − 1 . A1 (1)(z − 1)
(30)
Finding the delay D2 of a random (tagged) customer that arrives in the secondary queue is somewhat more involved. Given that the server is on vacation during its arrival slot, the tagged customer is served after all customers present in the system upon arrival of this customer are served and after all customers arriving in the primary queue during the remaining vacation time are served as depicted in Fig. 3, i.e., ∗
D2 = R + S +
U
Sj ,
(31)
j =1
where U ∗ = U1 + U2 + A1 + A∗2 +
R
A1,i
(32)
i=1
denotes the number of customers that are served before the tagged customer. Also, A1,i denotes the number of arrivals during the ith slot following the tagged customer’s arrival slot, whereas A1 and A∗2 denote the number of arrivals in the primary queue during the tagged customer’s arrival slot and the number of arrivals in the secondary queue prior to the tagged customer in this slot, respectively.
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Fig. 4. Delay for a tagged customer arriving in the secondary queue during a busy slot.
Given that the server is busy (see Fig. 4), the customer can only be served after the next vacation. Also, all customers present in the secondary queue upon arrival of the tagged customer as well as all customers arriving in the primary queue during the vacation are served before the tagged customer. Let U1∗ denote the number of customers in the primary queue at the beginning of the slot following the departure slot of the customer in service during the tagged customer’s arrival slot, then U1∗
= U1 + A1 − 1 +
H
A1,i .
(33)
i=1
Recall that the sub-busy period of a customer denotes the number of slots it takes to reduce the primary queue size with a single customer from the moment this customer starts service. Then, as the server only takes a vacation when the primary queue is empty, the vacation starts after a number of slots equal to the sum of the sub-busy periods of all U1∗ customers present in the primary queue at the beginning of the slot following the departure slot of the customer that is in service during the tagged customer’s arrival slot. The customer delay is then given by ∗
D2 = H +
U1
∗
Xj + V1 +
j =1
U2
Sˆj + S,
(34)
j =1
where U2∗ = U2 + A∗2 +
V1
Aˆ 1,i
(35)
i=1
denotes the number of customers that are served after the vacation but before the tagged customer. Further, Xj denotes the sub-busy period of the j th customer in the primary queue after departure of the customer in service during the tagged customer’s arrival slot, Aˆ 1,j denotes the number of primary arrivals in the
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j th vacation slot following the tagged customer’s arrival slot and Sˆj denotes the service time of the j th customer served after the vacation following the tagged customer’s arrival slot. The former equations then easily yield D2 (z) = M(zA1 (S(z)), S(z), S(z))A∗ (S(z), S(z))S(z) + N(zA1 (X(z)), X(z), S(z))A∗ (X(z), S(z))V1 (zA1 (S(z)))S(z)
(36)
in which A∗ (z1 , z2 ) denotes the joint probability generating function of the number of arrivals in the primary queue A1 and those in the secondary queue arriving prior to the tagged customer A∗2 (see, e.g., [9]) A∗ (z1 , z2 ) =
A(z1 , z2 ) − A1 (z1 ) . A2 (1)(z2 − 1)
(37)
The probability generating function of the sub-busy period X(z) of a customer in (36) is implicitly defined by X(z) = S(zA1 (X(z)))
(38)
as the sub-busy period of a customer equals the sum of its service time and of the sub-busy periods of all arrivals during its service time (see, e.g., [10]). 4. Numerical example We now consider the gated-exhaustive model in the particular case when both arrival streams are independent Poisson processes and when the service times are (shifted) geometrically distributed, i.e., z A(z1 , z2 ) = eλ1 (z1 −1)+λ2 (z2 −1) , S(z) = , (39) µB + (1 − µB )z where λ1 and λ2 denote the mean number of arrivals per slot in primary and secondary queues, respectively, and where µB denotes the mean customer service time. Further, vacation times are fixed to a constant L and we consider both the single and multiple vacation policies, i.e., V1 (z) = V2 (z) = zL ,
(40)
in case of multiple vacations, whereas V1 (z) = zL
and
V2 (z) = z
(41)
in case of single vacations. Fig. 5 depicts the primary (a) and secondary (b) queue contents vs. the fraction of customers that arrive in the primary queue x. Mean customer service time µB equals five slots and server vacations take L = 20 slots for all curves. We consider both the single- and multiple-vacation systems for different loads ρ as depicted. Clearly x = 0 corresponds to the purely gated vacation system whereas x = 1 corresponds to the purely exhaustive vacation system. As expected, the single-vacation system always outperforms the multiple-vacation system. Higher x imply less vacation periods as more customers arrive after the gate. Therefore, total queue contents decreases for increasing x. Further, secondary queue contents decreases
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Fig. 5. Mean queue contents vs. fraction x.
Fig. 6. Buffer correlation vs. fraction x (a) and mean delay vs. load (b).
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for increasing values of x as there are less arrivals in the secondary queue. Mean primary queue contents, however, either decreases or increases for increasing values of x. If the load is low or if the system operates nearly exhaustive, increasing x implies increasing mean primary queue contents. For more heavily loaded systems and smaller values of x, increasing x implies decreasing mean primary queue contents. Fig. 6(a) depicts the correlation factor ρU between primary and secondary queue contents at random slot boundaries. Again, mean customer service time µB equals five slots and vacations take L = 20 slots. We assume a multiple-vacation policy. The different curves correspond to different system loads ρ as depicted. The curves indicate a reasonable amount of correlation between both buffers and both positive and negative correlation is possible. Fig. 6(b) depicts mean customer delays µD1 and µD2 for customers arriving in primary and secondary queues (primary and secondary delay), respectively, vs. the total arrival load ρ for the multiple-vacation policy. Mean customer service time µB equals five slots whereas vacations take L = 20 slots. Different values for the fraction of customers that arrive in the primary queue x are assumed as depicted. Increasing load implies longer delays and the mean primary delay is shorter than the mean secondary delay, as expected. Further, mean secondary delay increases if there is more primary traffic. If the system load ρ is low, increasing the fraction of primary traffic, will also increase the primary delay. However, for more heavily loaded systems, the opposite is the case. 5. Conclusions We considered the combined gated-exhaustive vacation system in discrete-time using a probability generating functions approach. Results include various performance measures such as moments of the queue contents at random slot boundaries and moments of the customer delay for customer’s arriving in either the primary or secondary queues. These expressions are explicit once one determines K numerically. The vacation model under consideration allowed to consider both single and multiple vacation cases. References [1] B.T. Doshi, Queueing systems with vacations—a survey, Queueing Syst. 1 (1986) 29–66. [2] H. Takagi, Queueing Analysis, A Foundation of Performance Evaluation, Vol. 1, Vacation and Priority Systems, Part 1, Elsevier, Amsterdam, 1991. [3] D. Fiems, B. Steyaert, H. Bruneel, Discrete-time queues with general service times and general server interruptions, in: Internet Performance and Control of Network Systems, Proceedings of the SPIE, Vol. 4211, Boston, USA, November 6–7, 2000. [4] H. Takagi, A survey of queueing analysis of polling models, in: Proceedings of the Third IFIP International Conference on Data Communication Systems and their Performance, Rio de Janeiro, Brazil, June 22–25, 1987. [5] F.A. Van der Duyn Schouten, S.G. Vanneste, Maintenance optimization of a production system with buffer capacity, Eur. J. Oper. Res. 82 (1995) 232–338. [6] D. Towsley, J.K. Wolf, On the statistical analysis of queue lengths and waiting times for statistical multiplexers with ARQ retransmission schemes, IEEE Trans. Commun. COM-27 (4) (1979) 693–702. [7] D. Fiems, H. Bruneel, Discrete-time queueing systems with vacations governed by geometrically distributed timers, in: Proceedings of the Africom 2001, Fifth International Conference on Communication Systems, Cape Town, South Africa, May 2001. [8] S. Sumita, Performance analysis of interprocessor communications in an electronic switching system with distributed control, Perform. Eval. 9 (1988–1989) 83–91.
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[9] J. Walraevens, B. Steyaert, H. Bruneel, Delay Characteristics in discrete-time GI-G-1 queues with non-preemptive priority queueing discipline, Perform. Eval., in press. [10] H. Bruneel, Performance of discrete-time queuing systems, Comput. Oper. Res. 20 (1993) 303–320. Dieter Fiems was born in Ghent, Belgium, in 1973. He received his engineering degree at KAHO-St-Lieven and a post-graduate degree in computer science at Ghent University in 1997 and 1998, respectively. Since then, he is working as a Ph.D. student at the Department of Telecommunications and Information Processing at Ghent university, as a member of the SMACS research group. His main research interests include discrete-time queueing models and stochastic modeling of IP and ATM networks.
Stijn De Vuyst was born in Eeklo, Belgium, in 1973. He obtained the M.S. degree in electrical engineering from Ghent University, Belgium, in 1997. Since October 1998, he is a member of the SMACS Research Group, Department of Telecommunications and Information Processing, also at Ghent University. His main interest involves the study of discrete-time queueing models.
Herwig Bruneel was born in Zottegem, Belgium, in 1954. He received the M.S. degree in electrical engineering, the degree of Licentiate in computer science, and the Ph.D. degree in computer science in 1978, 1979 and 1984, respectively, all from Ghent University, Belgium. He is full time Professor in the Faculty of Applied Sciences and head of the Department of Telecommunications and Information Processing at the same university. He also leads the SMACS Research Group within this department. His main personal research interests include stochastic modeling and analysis of communication systems, discrete-time queueing theory, and the study of ARQ protocols. He has published more than 170 papers on these subjects and is a co-author of the book H. Bruneel and B.G. Kim, “Discrete-Time Models for Communication Systems Including ATM” (Kluwer Academic Publishers, Boston, 1993). Since October 2001, he serves as the Academic Director for Research Affairs at Ghent University.
The combined gated-exhaustive vacation system in discrete time Dieter Fiems∗ , Stijn De Vuyst1 , Herwig Bruneel2 SMACS Research Group, Vakgroep TELIN (TW07), Ghent University, St-Pietersnieuwstraat 41, B-9000 Gent, Belgium
Abstract We consider a discrete-time gated vacation system. The available buffer space is divided into two subsequent queues separated by a gate and new customers arrive either before or after this gate. Whenever all customers after the gate are served, the server takes a vacation. After each vacation, the gate opens which causes all waiting customers to move to the buffer space after the gate. The model under investigation allows to capture performance of a.o. the exhaustive and the gated queueing systems with multiple or single vacations. Using a probability generating functions approach, we obtain expressions for performance measures such as moments of system contents at various epochs in equilibrium and of customer delay. We conclude with a numerical example. © 2002 Elsevier Science B.V. All rights reserved. Keywords: Discrete-time queue; Vacation; Interruption
1. Introduction Queueing systems with server vacations [1,2] have proven to be a useful abstraction of systems where several classes of customers share a common resource such as priority systems [3] and polling systems [4], or of systems where this resource is unreliable such as maintenance models [5], ARQ systems [6], etc. Typically, one distinguishes following classes of vacation systems [2]: For exhaustive systems, the server only takes a vacation whenever there are no more customers in the system. The gated system keeps on serving until there are no customers left that arrived before the end of the last vacation. Number limited systems pose a maximum number of customers that can be served between server vacations whereas time-limited systems pose a maximal time between vacations, either in a preemptive or non-preemptive way [7]. For both number- and time-limited systems, the server also takes a vacation whenever the queue empties before the respective maxima expire. Additionally, one distinguishes multiple vacation systems—the server takes another vacation if there are no customers in the system when it returns from a vacation—and single vacation systems—the server waits for the first arrival when there are no customers in the system upon returning from a vacation. ∗ Corresponding author. Tel.: +32-9-264-8902; fax: +32-9-264-4295. E-mail addresses: [email protected] (D. Fiems), [email protected] (S. De Vuyst), [email protected] (H. Bruneel). 1 Tel.: +32-9-264-8902; fax: +32-9-264-4295. 2 Tel.: +32-9-264-3414; fax: +32-9-264-4295.
0166-5316/02/$ – see front matter © 2002 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 6 - 5 3 1 6 ( 0 2 ) 0 0 1 1 1 - 6
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The present contribution uses the method of the supplementary variable to investigate a model combining all features of both the gated and the exhaustive vacation system. The system takes multiple vacations and the first of these has an alternative distribution [8]. The model under investigation therefore allows to capture performance for both the single-and the multiple-vacation cases. The rest of the paper is organized as follows. Section 2 gives a detailed description of our combined gated-exhaustive vacation system. The performance analysis is presented and numerically illustrated in Sections 3 and 4, respectively, whereas conclusions are drawn in Section 5. 2. Mathematical model We consider a discrete-time queueing system, i.e., time is divided into fixed length intervals or slots. The system consists of a single server and two infinite capacity queues separated by a gate as depicted in Fig. 1. During the consecutive slots, customers arrive either in the primary or in the secondary queues. Customers in both queues are served according to a first-in first-out (FIFO) service discipline. The customers in the secondary queue move to the primary queue whenever the gate opens (see further). The numbers of customers arriving in the primary and secondary queues during the consecutive slots are modeled by means of an independent and identically distributed (i.i.d.) series of jointly dependent random variables with common probability mass function a(m, n) (m, n ≥ 0) and corresponding probability generating function A(z1 , z2 ). For ease of notation, we also introduce the generating function A(z) = A(z, z) of the total number of arrivals per slot and the marginal probability generating functions A1 (z) = A(z, 1) and A2 (z) = A(1, z) corresponding to the number of arrivals per slot in the primary and secondary queues, respectively. Service of customers is synchronized with respect to slot boundaries, which implies that service of a customer cannot start before the slot following its arrival slot. Service typically takes a number of slots and the service times (in slots) of the consecutive customers are modeled by means of an i.i.d. series of positive random variables with common probability mass function s(n) (n > 0) and with corresponding probability generating function S(z). Furthermore, we shall refer to an uninterrupted sequence of service times as a busy period. From these definitions, it is clear that the mean number of arrivals per slot and the average amount of work (in terms of slots) per arriving customer is given by A (1) and S (1), respectively, due to the moment-generating property of probability generating functions. Hence, the overall workload ρ is equal to A (1)S (1) and since there is only one server, the stability condition for this system is given by ρ A (1)S (1) < 1.
(1)
Whenever there are no more primary customers in the system at the beginning of a slot, the server takes a vacation, meaning that for an integer number of contiguous slots no customers will be served. Upon
Fig. 1. Gated-exhaustive queueing system.
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returning from this vacation the gate opens, i.e., all customers in the secondary queue move to the primary at the end of the last vacation slot. If there are no customers in the system upon returning from the vacation, the server immediately takes a new vacation. The lengths (in slots) of the consecutive server vacations are modeled as a series of independent random variables with common probability generating functions V1 (z) or V2 (z) depending on whether the vacation is not or is immediately preceded by another vacation. As with the service times, we refer to an uninterrupted sequence of vacations as a vacation period. As such, the system continuously alternates between busy periods and vacation periods. Clearly, this model encapsulates the gated—only arrivals in secondary queue—and exhaustive—only arrivals in the primary queue—vacation systems with multiple (V1 (z) = V2 (z)) and single (V2 (z) = z) vacations. Further, the model under consideration also relates to priority systems as customers arriving in the primary queue receive priority over customers arriving in the secondary queue. However, if there are no more customers in the primary queue, waiting (low priority) customers in the secondary queue can jump to the (high priority) primary queue.
3. Queueing analysis 3.1. System state at random slot boundaries Let U1,k and U2,k , respectively, denote the number of customers in the primary and secondary queues at the beginning of the kth slot. Further, let Rk denote the remaining number of vacation slots following slot k if slot k is a vacation slot and let Hk denote the number of remaining service slots of the customer in service following slot k if this is not the case. Also, for ease of analysis, we introduce an additional random variable Tk , indicating whether the server is on vacation (Tk = V ) or busy (Tk = B) during slot k. The system under consideration then yields the following set of system equations: • Given Tk = V and Rk > 0, then, during slot (k + 1), the server continues its vacation: Tk+1 = V ,
Rk+1 = Rk − 1,
U1,k+1 = U1,k + A1,k ,
U2,k+1 = U2,k + A2,k .
(2)
• Given Tk = V , Rk = 0 and U1,k + U2,k + A1,k + A2,k = 0, then, slot k is the last slot of a vacation after which the system is found to be empty. As such, the server immediately takes a new vacation: Tk+1 = V ,
Rk+1 = V2 − 1,
U1,k+1 = 0,
U2,k+1 = 0.
(3)
• Given Tk = V , Rk = 0 and U1,k + U2,k + A1,k + A2,k > 0, then, the server returns from a vacation and finds the system non-empty. The gate opens and a new customer starts service: Tk+1 = B,
Hk+1 = S − 1,
U1,k+1 = U1,k + U2,k + A1,k + A2,k ,
U2,k+1 = 0.
(4)
• Given Tk = B and Hk > 0, then, the customer in service continues service in slot (k + 1): Tk+1 = B,
Hk+1 = Hk − 1,
U1,k+1 = U1,k + A1,k ,
U2,k+1 = U2,k + A2,k .
(5)
• Given Tk = B, Hk = 0 and U1,k + A1,k > 1, then, the customer in service leaves the system and a new customer starts service: Tk+1 = B,
Hk+1 = S − 1,
U1,k+1 = U1,k + A1,k − 1,
U2,k+1 = U2,k + A2,k .
(6)
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• Given Tk = B, Hk = 0, U1,k = 1 and A1,k = 0, then, the current busy period terminates as there are no customers left in the primary queue at the beginning of slot (k + 1). The server therefore starts a vacation: Tk+1 = V ,
Rk+1 = V1 − 1,
U1,k+1 = 0,
U2,k+1 = U2,k + A2,k .
(7)
Here A1,k and A2,k denote the number of arrivals during slot k in the primary and secondary queues, respectively, S denotes the service time of a random customer, and V1 and V2 denote the lengths of server vacations not immediately or immediately preceded by another vacation. It can be seen from the above system equations that the vector {U1,k , U2,k , Rk or Hk , Tk } forms a four-dimensional Markov chain, where the variable Rk only has meaning if Tk = V and Hk only if Tk = B. As such, this vector provides a complete description for the state of the system at any given slot k. In what follows, we show how to obtain the equilibrium distribution of the system state, both during busy slots and vacation slots. Now, let Nk (x, z1 , z2 ) and Mk (x, z1 , z2 ) denote the following (partial) joint probability generating functions U
−1 U2,k z2 |Tk = B] Pr[Tk = B], Rk U1,k U2,k E[x z1 z2 |Tk = V ] Pr[Tk = V ].
Nk (x, z1 , z2 ) = E[x Hk z1 1,k Mk (x, z1 , z2 ) =
(8)
The preceding system equations (2)–(7) can be used to relate these functions for slot (k + 1) to those for slot k. After some standard z-transform manipulations, we find Nk+1 (x, z1 , z2 ) =
S(x) (Mk (0, z1 , z1 )A(z1 ) − Mk (0, 0, 0)A(0)) xz1 S(x) + (Nk (0, z1 , z2 )A(z1 , z2 ) − Nk (0, 0, z2 )A(0, z2 )) xz1 A(z1 , z2 ) + (Nk (x, z1 , z2 ) − Nk (0, z1 , z2 )), x
(9)
and A(z1 , z2 ) V2 (x) (Mk (x, z1 , z2 ) − Mk (0, z1 , z2 )) + A(0)Mk (0, 0, 0) x x V1 (x) + A(0, z2 )Nk (0, 0, z2 ). (10) x Under the assumption (1) that the system reaches equilibrium, let N(x, z1 , z2 ) and M(x, z1 , z2 ) denote the equilibrium probability generating functions, i.e., N(x, z1 , z2 ) = limk→∞ Nk (x, z1 , z2 ) and M(x, z1 , z2 ) = limk→∞ Mk (x, z1 , z2 ). Letting k → ∞ in (9) and (10), and solving for N(x, z1 , z2 ) and M(x, z1 , z2 ), respectively, then yields Mk+1 (x, z1 , z2 ) =
S(x)M(0, z1 , z1 )A(z1 ) − S(x)A(0, z2 )N(0, 0, z2 ) −A(z1 , z2 )(z1 − S(x))N(0, z1 , z2 ) − S(x)M(0, 0, 0)A(0) N(x, z1 , z2 ) = (x − A(z1 , z2 ))z1
(11)
and M(x, z1 , z2 ) =
A(0)V2 (x)M(0, 0, 0) − A(z1 , z2 )M(0, z1 , z2 ) + A(0, z2 )V1 (x)N(0, 0, z2 ) . x − A(z1 , z2 )
(12)
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Clearly, as both generating functions are analytic within the unit disk, the numerator on the right-hand side of both equations (11) and (12) vanishes for (A(z1 , z2 ), z1 , z2 ), implying N(0, z1 , z2 ) = S(A(z1 , z2 ))
A(z1 )M(0, z1 , z1 ) − A(0)M(0, 0, 0) − A(0, z2 )N(0, 0, z2 ) A(z1 , z2 )(z1 − S(A(z1 , z2 )))
(13)
and M(0, z1 , z2 ) =
A(0)V2 (A(z1 , z2 ))M(0, 0, 0) + A(0, z2 )V1 (A(z1 , z2 ))N(0, 0, z2 ) . A(z1 , z2 )
(14)
From Eqs. (11)–(14) it is clear that M(x, z1 , z2 ) and N(x, z1 , z2 ) are completely determined once one determines M(0, 0, 0) and N(0, 0, z). The quantity M(0, 0, 0) can be directly obtained from the normalization condition N(1, 1, 1) + M(1, 1, 1) = 1. Using de l’Hˆopital’s rule, this yields M(0, 0, 0) =
1 − S (1)A (1) − V1 (1)A1 (0)N(0, 0, 1) . A(0)V2 (1)
(15)
One may also obtain the former expression as follows. The fraction of busy slots is given by the load S (1)A (1). As the system is either busy or on vacation, the fraction of vacation slots is given by 1 − S (1)A (1). On the other hand, M(0, 0, 0)A(0) and N(0, 0, 1)A1 (0) represent the probabilities that a new vacation starts at the end of a random vacation and busy slot, respectively. The fraction of vacation slots is therefore also given by N(0, 0, 1)A1 (0)V1 (1) + M(0, 0, 0)A(0)V2 (1). Comparison of these fractions then yields the former expression for M(0, 0, 0). Finally, to characterize the last unknown function N(0, 0, z), we can proceed as follows. First notice that for each |z2 | ≤ 1, Rouché’s theorem assures the existence of a unique value χ (z2 ) such that, |χ (z2 )| ≤ 1 and χ(z2 ) = S(A(χ(z2 ), z2 )).
(16)
As the partial probability generating function N(0, z1 , z2 ) is analytic for |z1 |, |z2 | ≤ 1, the numerator on the right-hand side of (13) vanishes for all values (0, χ(z2 ), z2 ), |z2 | ≤ 1. As an alternative to N(0, 0, z), let Q(z) denote Q(z)
A(0, z)N(0, 0, z) , A1 (0)N(0, 0, 1)
(17)
i.e., Q(z) is the probability generating function of the number of customers in the system at the end of a busy period. Eqs. (13) and (14) then yield the following functional equation for Q(z): Q(χ(z)) =
Q(z) + K(1 − V2 (A(χ(z))) V1 (A(χ(z)))
(18)
with K
A(0)M(0, 0, 0) . A1 (0)N(0, 0, 1)
(19)
Now, consider the series, x0 = 0, xi = χ (xi−1 ) for i > 0. This series clearly converges to 1 as long as χ (1) < 1 which is implied by S (1)A (1) < 1. The latter condition is always met if the system reaches equilibrium due to (1). The convergence is brought about by the fact that χ (z) proves to be a probability
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generating function. In particular, it is the probability generating function of the number of secondary customers that arrive during the sub-busy period of a primary customer (see e.g., [9]). The sub-busy period of a primary customer is defined to start from the moment this customer enters service, and ends on the first occasion in which the (primary) queue contains one customer less than at the beginning of the sub-busy period. The concept of sub-busy periods will be further exploited in the next paragraphs when studying delay times. As the series {xi } converges to 1, the series yi = K/Q(xi ) converges to K since Q(1) = 1. From Eqs. (14), (17) and (19) one easily obtains M(0, 0, 0) K V1 (A(0)) y0 = = , (20) Q(0) N(0, 0, 0) 1 − V2 (A(0)) whereas, Eq. (18) yields V1 (A(xi+1 ))yi yi+1 = . 1 + (1 − V2 (A(xi+1 )))yi
(21)
These relations then allow to calculate K numerically up to any desired precision. Note that—as in [8]— by iteration of (18), one may also obtain an explicit expression for Q(z). The unknown value K then follows from the normalization condition Q(1) = 1. Summarizing, we get Q(z2 )(V1 (x) − V1 (A(z1 , z2 )) + K(V2 (x) − V2 (A(z1 , z2 )))) M(x, z1 , z2 ) = c (22) x − A(z1 , z2 ) and N(x, z1 , z2 ) = c
S(x) − S(A(z1 , z2 )) V1 (A(z1 ))Q(z1 ) − Q(z2 ) − K(1 − V2 (A(z1 ))) z1 − S(A(z1 , z2 )) x − A(z1 , z2 )
(23)
with c=
1 − A (1)S (1) , V1 (1) + KV2 (1)
(24)
where Q(z) is implicitly defined by (18) and where one can determine K numerically using Eqs. (20) and (21). 3.2. Performance measures: queue contents and customer delay We now show how the partial probability generating functions M(x, z1 , z2 ) and N(x, z1 , z2 ) allow to evaluate the performance of the system. Let U (z1 , z2 ) denote the joint probability generating function of the number of customers in the primary and secondary queues at random slot boundaries, then, using the definitions (8), we get U (z1 , z2 ) = M(1, z1 , z2 ) + z1 N(1, z1 , z2 ).
(25)
Analogously, the probability generating function V (z1 , z2 ) of the number of customers in primary and secondary queues at departure epochs—the beginning of a slot following a slot where a customer leaves the system—is given by N(0, z1 , z2 ) V (z1 , z2 ) = (26) A(z1 , z2 ). N(0, 1, 1)
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Fig. 2. Delay experienced by an arbitrary but tagged customer arriving in the primary queue either (a) during a vacation slot or (b) a busy slot. Customers that are served during the delay time of the tagged customer are shaded.
Using the moment-generating property of probability generating functions, one easily obtains performance measures such as mean and variance of the number of customers in primary and secondary queues and such as the correlation between the number of customers in primary and secondary queues. Further let D1 and D2 denote the delay—the number of slots between the end of a customer’s arrival slot and the end of its departure slot—of a random customer that arrives in the primary and secondary queues, respectively, and let D1 (z) and D2 (z) denote the corresponding probability generating functions. Also, let U1 and U2 denote the primary and secondary queue contents at the beginning of a random customer’s arrival slot and let H (R) denote the remaining service (vacation) time given that the server is busy (on vacation) during the random customer’s arrival slot. We first consider a random “but tagged” customer that arrives in the primary queue. Such a customer receives service when all customers that arrived in the primary queue before the tagged customer are served as depicted in Fig. 2. Given that the customer under consideration arrives during a vacation slot, its delay equals the remaining vacation time, augmented by the sum of the service times of the customers present in the primary queue upon arrival of the tagged customer and by its own service time, i.e., U1 +A∗1
D1 = R + S +
Sj ,
(27)
j =1
where A∗1 denotes the number of arrivals in the primary queue during the tagged customer’s arrival slot prior to this customer. Further, Sj denotes the service time of the j th customer starting service after the tagged customer’s arrival slot and S denotes the tagged customer’s own service time. Similarly, given that the customer arrives during a busy slot, its delay equals the remaining service time of the customer in service upon arrival of the tagged customer, augmented by the sum of the service times of the customers present in the primary queue upon arrival of the tagged customer and by its own service time, i.e., U1 +A∗1 −1
D1 = H + S +
j =1
Sj .
(28)
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Fig. 3. Delay for a tagged customer arriving in the secondary queue during a vacation slot.
Due to the independent nature of the arrival process, the state of the system at the beginning of a random customer’s arrival slot has the same stochastical properties as the state of the system during a random slot and therefore, D1 (z) = (M(z, S(z), 1) + N(z, S(z), 1))A∗1 (S(z))S(z),
(29)
where A∗1 (z) denotes the probability generating function corresponding to A∗1 (see, e.g., [10]), A∗1 (z) =
A1 (z) − 1 . A1 (1)(z − 1)
(30)
Finding the delay D2 of a random (tagged) customer that arrives in the secondary queue is somewhat more involved. Given that the server is on vacation during its arrival slot, the tagged customer is served after all customers present in the system upon arrival of this customer are served and after all customers arriving in the primary queue during the remaining vacation time are served as depicted in Fig. 3, i.e., ∗
D2 = R + S +
U
Sj ,
(31)
j =1
where U ∗ = U1 + U2 + A1 + A∗2 +
R
A1,i
(32)
i=1
denotes the number of customers that are served before the tagged customer. Also, A1,i denotes the number of arrivals during the ith slot following the tagged customer’s arrival slot, whereas A1 and A∗2 denote the number of arrivals in the primary queue during the tagged customer’s arrival slot and the number of arrivals in the secondary queue prior to the tagged customer in this slot, respectively.
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Fig. 4. Delay for a tagged customer arriving in the secondary queue during a busy slot.
Given that the server is busy (see Fig. 4), the customer can only be served after the next vacation. Also, all customers present in the secondary queue upon arrival of the tagged customer as well as all customers arriving in the primary queue during the vacation are served before the tagged customer. Let U1∗ denote the number of customers in the primary queue at the beginning of the slot following the departure slot of the customer in service during the tagged customer’s arrival slot, then U1∗
= U1 + A1 − 1 +
H
A1,i .
(33)
i=1
Recall that the sub-busy period of a customer denotes the number of slots it takes to reduce the primary queue size with a single customer from the moment this customer starts service. Then, as the server only takes a vacation when the primary queue is empty, the vacation starts after a number of slots equal to the sum of the sub-busy periods of all U1∗ customers present in the primary queue at the beginning of the slot following the departure slot of the customer that is in service during the tagged customer’s arrival slot. The customer delay is then given by ∗
D2 = H +
U1
∗
Xj + V1 +
j =1
U2
Sˆj + S,
(34)
j =1
where U2∗ = U2 + A∗2 +
V1
Aˆ 1,i
(35)
i=1
denotes the number of customers that are served after the vacation but before the tagged customer. Further, Xj denotes the sub-busy period of the j th customer in the primary queue after departure of the customer in service during the tagged customer’s arrival slot, Aˆ 1,j denotes the number of primary arrivals in the
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j th vacation slot following the tagged customer’s arrival slot and Sˆj denotes the service time of the j th customer served after the vacation following the tagged customer’s arrival slot. The former equations then easily yield D2 (z) = M(zA1 (S(z)), S(z), S(z))A∗ (S(z), S(z))S(z) + N(zA1 (X(z)), X(z), S(z))A∗ (X(z), S(z))V1 (zA1 (S(z)))S(z)
(36)
in which A∗ (z1 , z2 ) denotes the joint probability generating function of the number of arrivals in the primary queue A1 and those in the secondary queue arriving prior to the tagged customer A∗2 (see, e.g., [9]) A∗ (z1 , z2 ) =
A(z1 , z2 ) − A1 (z1 ) . A2 (1)(z2 − 1)
(37)
The probability generating function of the sub-busy period X(z) of a customer in (36) is implicitly defined by X(z) = S(zA1 (X(z)))
(38)
as the sub-busy period of a customer equals the sum of its service time and of the sub-busy periods of all arrivals during its service time (see, e.g., [10]). 4. Numerical example We now consider the gated-exhaustive model in the particular case when both arrival streams are independent Poisson processes and when the service times are (shifted) geometrically distributed, i.e., z A(z1 , z2 ) = eλ1 (z1 −1)+λ2 (z2 −1) , S(z) = , (39) µB + (1 − µB )z where λ1 and λ2 denote the mean number of arrivals per slot in primary and secondary queues, respectively, and where µB denotes the mean customer service time. Further, vacation times are fixed to a constant L and we consider both the single and multiple vacation policies, i.e., V1 (z) = V2 (z) = zL ,
(40)
in case of multiple vacations, whereas V1 (z) = zL
and
V2 (z) = z
(41)
in case of single vacations. Fig. 5 depicts the primary (a) and secondary (b) queue contents vs. the fraction of customers that arrive in the primary queue x. Mean customer service time µB equals five slots and server vacations take L = 20 slots for all curves. We consider both the single- and multiple-vacation systems for different loads ρ as depicted. Clearly x = 0 corresponds to the purely gated vacation system whereas x = 1 corresponds to the purely exhaustive vacation system. As expected, the single-vacation system always outperforms the multiple-vacation system. Higher x imply less vacation periods as more customers arrive after the gate. Therefore, total queue contents decreases for increasing x. Further, secondary queue contents decreases
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Fig. 5. Mean queue contents vs. fraction x.
Fig. 6. Buffer correlation vs. fraction x (a) and mean delay vs. load (b).
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for increasing values of x as there are less arrivals in the secondary queue. Mean primary queue contents, however, either decreases or increases for increasing values of x. If the load is low or if the system operates nearly exhaustive, increasing x implies increasing mean primary queue contents. For more heavily loaded systems and smaller values of x, increasing x implies decreasing mean primary queue contents. Fig. 6(a) depicts the correlation factor ρU between primary and secondary queue contents at random slot boundaries. Again, mean customer service time µB equals five slots and vacations take L = 20 slots. We assume a multiple-vacation policy. The different curves correspond to different system loads ρ as depicted. The curves indicate a reasonable amount of correlation between both buffers and both positive and negative correlation is possible. Fig. 6(b) depicts mean customer delays µD1 and µD2 for customers arriving in primary and secondary queues (primary and secondary delay), respectively, vs. the total arrival load ρ for the multiple-vacation policy. Mean customer service time µB equals five slots whereas vacations take L = 20 slots. Different values for the fraction of customers that arrive in the primary queue x are assumed as depicted. Increasing load implies longer delays and the mean primary delay is shorter than the mean secondary delay, as expected. Further, mean secondary delay increases if there is more primary traffic. If the system load ρ is low, increasing the fraction of primary traffic, will also increase the primary delay. However, for more heavily loaded systems, the opposite is the case. 5. Conclusions We considered the combined gated-exhaustive vacation system in discrete-time using a probability generating functions approach. Results include various performance measures such as moments of the queue contents at random slot boundaries and moments of the customer delay for customer’s arriving in either the primary or secondary queues. These expressions are explicit once one determines K numerically. The vacation model under consideration allowed to consider both single and multiple vacation cases. References [1] B.T. Doshi, Queueing systems with vacations—a survey, Queueing Syst. 1 (1986) 29–66. [2] H. Takagi, Queueing Analysis, A Foundation of Performance Evaluation, Vol. 1, Vacation and Priority Systems, Part 1, Elsevier, Amsterdam, 1991. [3] D. Fiems, B. Steyaert, H. Bruneel, Discrete-time queues with general service times and general server interruptions, in: Internet Performance and Control of Network Systems, Proceedings of the SPIE, Vol. 4211, Boston, USA, November 6–7, 2000. [4] H. Takagi, A survey of queueing analysis of polling models, in: Proceedings of the Third IFIP International Conference on Data Communication Systems and their Performance, Rio de Janeiro, Brazil, June 22–25, 1987. [5] F.A. Van der Duyn Schouten, S.G. Vanneste, Maintenance optimization of a production system with buffer capacity, Eur. J. Oper. Res. 82 (1995) 232–338. [6] D. Towsley, J.K. Wolf, On the statistical analysis of queue lengths and waiting times for statistical multiplexers with ARQ retransmission schemes, IEEE Trans. Commun. COM-27 (4) (1979) 693–702. [7] D. Fiems, H. Bruneel, Discrete-time queueing systems with vacations governed by geometrically distributed timers, in: Proceedings of the Africom 2001, Fifth International Conference on Communication Systems, Cape Town, South Africa, May 2001. [8] S. Sumita, Performance analysis of interprocessor communications in an electronic switching system with distributed control, Perform. Eval. 9 (1988–1989) 83–91.
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[9] J. Walraevens, B. Steyaert, H. Bruneel, Delay Characteristics in discrete-time GI-G-1 queues with non-preemptive priority queueing discipline, Perform. Eval., in press. [10] H. Bruneel, Performance of discrete-time queuing systems, Comput. Oper. Res. 20 (1993) 303–320. Dieter Fiems was born in Ghent, Belgium, in 1973. He received his engineering degree at KAHO-St-Lieven and a post-graduate degree in computer science at Ghent University in 1997 and 1998, respectively. Since then, he is working as a Ph.D. student at the Department of Telecommunications and Information Processing at Ghent university, as a member of the SMACS research group. His main research interests include discrete-time queueing models and stochastic modeling of IP and ATM networks.
Stijn De Vuyst was born in Eeklo, Belgium, in 1973. He obtained the M.S. degree in electrical engineering from Ghent University, Belgium, in 1997. Since October 1998, he is a member of the SMACS Research Group, Department of Telecommunications and Information Processing, also at Ghent University. His main interest involves the study of discrete-time queueing models.
Herwig Bruneel was born in Zottegem, Belgium, in 1954. He received the M.S. degree in electrical engineering, the degree of Licentiate in computer science, and the Ph.D. degree in computer science in 1978, 1979 and 1984, respectively, all from Ghent University, Belgium. He is full time Professor in the Faculty of Applied Sciences and head of the Department of Telecommunications and Information Processing at the same university. He also leads the SMACS Research Group within this department. His main personal research interests include stochastic modeling and analysis of communication systems, discrete-time queueing theory, and the study of ARQ protocols. He has published more than 170 papers on these subjects and is a co-author of the book H. Bruneel and B.G. Kim, “Discrete-Time Models for Communication Systems Including ATM” (Kluwer Academic Publishers, Boston, 1993). Since October 2001, he serves as the Academic Director for Research Affairs at Ghent University.