- Email: [email protected]
WV)
Performance Evaluation 50 (2002) 41–52
M/M/1 queues with working vacations (M/M/1/WV)夽 L.D. Servi∗ , S.G. Finn Lincoln Laboratory, Massachusetts Institute of Technology, 244 Wood Street, Lexington, MA 02420-9180, USA Received 5 June 2001; received in revised form 25 January 2002
Abstract The classical single server vacation model is generalized to consider a server which works at a different rate rather than completely stops during the vacation period. Simple explicit formulae for the mean, variance, and distribution of the number and time in the system are presented. The distributional results generalize the classical vacation decomposition with no service during a vacation. This model approximates a multi-queue system whose service rate is one of the two speeds for which the fast speed mode cyclically moves from queue to queue with an exhaustive schedule. This work is motivated and illustrated by the analysis of a WDM optical access network using multiple wavelengths which can be reconfigured. © 2002 Elsevier Science B.V. All rights reserved. Keywords: M/M/1 queues; Vacation model; Working vacation; WDM optical access
1. A single queue working vacation system Consider a single server queue which begins a working vacation of mean duration 1/η when the system is empty. If the system is empty when a working vacation ends, another begins. During the working vacation customers are served at a mean rate of µV . When the server is not on a working vacation customers are served at a mean rate of µB . The interarrival times, service times and vacation times are all exponentially distributed. Let N be the number in the system, λ the arrival rate, 1/η the average duration of a working vacation, 1/µB the average service time when the server is not on a working vacation, 1/µV the average service time when the server is on a working vacation, pB (j ) the probability of j in the system when the server is not on a working vacation, pj V (j ) the probability of j in the system when the server is on working vacation, Pi (z) = ∞ j =0 pi (j )z for i = B or V , and P (z) = PB (z) + PV (z). The following theorem uses classical methods to prove that N plus a geometrical distributed random variable equals the number in the system in the absence of vacations plus another geometrically distributed 夽 This work is sponsored by DARPA under contract F19628-00-C-0002. Opinions, interpretations, recommendations and conclusions are those of the authors and are not necessarily endorsed by the Department of Defense. ∗ Corresponding author. E-mail addresses: [email protected] (L.D. Servi), [email protected] (S.G. Finn).
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 0 5 7 - 3
42
L.D. Servi, S.G. Finn / Performance Evaluation 50 (2002) 41–52
random variable, or, equivalently, N is the weighted average of two geometrical distributions. If µV = 0 these results specialize to the known decomposition for M/M/1 systems with exponential vacations. Theorem. The number in the system for an M/M/1 with exponential working vacation has an z transform 1 − λ/µB 1 − zˆ −1 1 − µV /ˆzµB −1 P (z) = 1 − (λ/µB )z 1 − zˆ −1 z 1 − (µV /ˆzµB )z 1 − λ/µB 1 − zˆ −1 =γ + (1 − γ ) , (1.1) 1 − (λ/µB )z 1 − zˆ −1 z a distribution
Pr[N = j ] = γ
1−
λ µB
λ µB
j
+ (1 − γ )(1 − zˆ −1 )(ˆz)−j ,
(1.2)
a mean E[N] =
λ 1 µV λ 1 + − =γ + (1 − γ ) , µB − λ zˆ − 1 zˆ µB − µV µB − λ zˆ − 1
(1.3)
and a variance Var[N] =
zˆ µV µB zˆ λµB + − , 2 2 (µB − λ) (ˆz − 1) (µB zˆ − µV )2
where (ˆz − 1)(λˆz − µV )µB γ = (µB zˆ − µV )(λˆz − µB )
and
zˆ =
λ + µV + η +
(1.4)
(λ + µV + η)2 − 4λµV . 2λ
(1.5)
Proof. See Appendix A.
Three cases are considered which specialize to previously known results: (i) If the system never takes a vacation, i.e., µB = µV or if η (and hence zˆ ) approaches ∞, from (1.5), γ = 1 so, from (1.1) N is a geometrical distribution with a rate λ/µB ; (ii) if the system is always on a working vacation, i.e., η = 0, then from (1.5), zˆ = µV /λ and γ = 0. Hence, from (1.1), N is a geometrical distribution with a rate 1/ˆz = λ/µV ; (iii) if the server does no work during a working vacation, i.e., µV = 0 then from (1.5), zˆ = (λ + η)/λ. Hence, from the first equation in (1.1), 1 − λ/µB η P (z) = , 1 − (λ/µB )z η + λ − zλ which is the classical vacation decomposition. From (A.7), Pr[N = 0|i = V ] = 1 −
Pr[N = 0 and i = V ] pV (0) =1− = zˆ −1 , Pr[i = V ] PV (1)
(1.6)
where i ∈ {B, V } is the state variable denoting whether the system is in a regular busy period or in a working vacation. Eq. (1.6) provides an interpretation of zˆ as well as is useful in Section 2.
L.D. Servi, S.G. Finn / Performance Evaluation 50 (2002) 41–52
43
Fig. 1. Distribution for the number in system for a working vacation model λ = 2, µB = 3.5, µV = 1.5, η = 0.05 and j = 0, . . . , 50.
The second equality in (1.1) is useful to compute (1.2). However, note that none of the analysis guarantees that 0 ≤ γ ≤ 1 (even if the system is stable). For example, if λ = 2, µB = 3.5, µV = 1.5, and η = 0.05 then γ ≈ −0.06, but as illustrated in Fig. 1, N has a well-behaved distribution with a mean of approximately 12.9. Corollary. The total time in the system for an M/M/1 with exponential working vacations has a Laplace transform −1 µB − λ λˆz − λ λˆzµB /µV − λ αw (s) = , (1.7) s + µB − λ s + λˆz − λ s + λˆzµB /µV − λ a mean E[W ] =
1 µV 1 + − , µB − λ λ(ˆz − 1) λ(ˆzµB − µV )
(1.8)
and a variance Var[W ] =
1 µ2V 1 + − . (µB − λ)2 λ2 (ˆz − 1)2 λ2 (µB zˆ − µV )2
(1.9)
Proof. If αw (s) is the Laplace transform of the time in the system, then Keilson and Servi [1] states four
44
L.D. Servi, S.G. Finn / Performance Evaluation 50 (2002) 41–52
necessary conditions for P (z) = αw (λ − λz).
(1.10)
(a) The arrivals are Poisson; (b) all arriving customers enter the system, and remain in the system until served; (c) the customers leave the system one at a time in the order of arrivals; and (d) for any time t, the arrival process after time t and the time in system of any customer arriving before t are independent. Since these four conditions clearly hold, (1.10) and therefore, using (1.1), (1.7) follows. Taking the derivative of (1.10) and evaluating at z = 0 leads to the classical Little’s law formula, E[W ] = E[N]/λ, and hence, from (1.3), (1.8) follows. Taking a second derivative of (1.10) and evaluating at z = 0, leads to Var[W ] = (Var[N] − E[N ])/λ2 so, using (1.2) and (1.3), (1.9) follows. 2. Application to multiple queue systems Consider the following n-queue generalization of a cyclic service queue model which is motivated by a reconfigurable WDM optical access network: a single token cyclically visits each queue. Each queue operates either at a fast service rate or a nominal service rate. A queue operates at its fast service rate as soon as it acquires the token. If queue i possesses the token (and hence is operating at its fast speed) and becomes empty, its service rate is instantly reduced to its nominal rate for future arrivals but the token requires a reconfiguration time to pass before it visits queue (i + 1) mod(n). This cycle continually repeats itself. Following previous cyclic service queue analysis, e.g., [3,4], one notes that from the point of view of queue i the server follows a working vacation model. Hence, the previous analysis can be applied if the working vacation time corresponding to queue i can be determined. However, to do so implicitly introduces the following two approximating assumptions: (i) the duration of the working vacation of queue i is exponentially distributed, and (ii) the duration of the working vacation is an independent random variable uncorrelated with the queue i’s busy period. These approximations were previously used for a related classical cyclic service queues (e.g., [3]) with some success although under very high load the impact of ignoring correlations can be problematic. To proceed, let λi be the arrival rate to queue i, µiB be the fast service rate of queue i, i.e., the rate when it possesses the token, µiV the nominal service rate at queue i, V i the average duration that queue i is operating with its nominal service rate mode, i.e., the working vacation duration, V i = 1/ηi , zˆ i the probability of queue i not empty given the token is not at queue i, cf. (1.5) and (1.6), C the average cycle time, i.e., the time between token visits to queue 1, B i the average busy period, i.e., the average time the token is at queue i. Note that on a given cycle this could equal zero. And ∆ be the average reconfiguration time, i.e., the time needed to switch from a nominal rate to a fast rate (it is assumed that switching from a fast to a nominal rate is instantaneous). If the system is stable, the average number of arrivals to queue i in a cycle must equal the average number of service completions during a cycle. Hence, for i = 0, . . . , n − 1, λi C = µiB B i + µiV (C − B i ) Pr[not idle|in a working vacation period]. From (1.6), this equals µiB B i + µiV (C − B i )/ˆzi so B i = φ i C,
(2.1)
i
where φ is defined by φi =
λi zˆ i − µiV . µiB zˆ i − µiV
(2.2)
L.D. Servi, S.G. Finn / Performance Evaluation 50 (2002) 41–52
45
The average cycle time, C, equals the total reconfiguration time plus the total time spent in the fast service time speed at each queue, i.e., C = n∆ +
n
Bj .
(2.3)
j =1
Combining (2.1) and (2.3) leads to C=
1−
n∆ n
j =1
φj
.
(2.4)
Finally, the average working vacation for queue i is set equal to the average time it is not working at the fast service time speed, i.e., 1 = V i = C − Bi. ηi
(2.5)
From (2.1), (2.4) and (2.5), 1 − nj=1 φ j i η = . n∆(1 − φ i )
(2.6)
As derived in Appendix A, the φ i ’s are a solution to ai (φ i )2 + bi φ i + ci φ i φˆ + di + ei φˆ = 0,
i = 1, . . . , n,
(2.7) n
where ai , bi , ci , di , and ei are given in (A.15)–(A.19) and φˆ = j =1 φ j . Eq. (2.7) can be solved by ˆ solving the n, now independent, quadratic equations in terms iteratively selecting an initial value of φ, ˆ The solution of φˆ and then φ i , i = 1, . . . , n of φˆ and then finding and selecting an improved value of φ. can be used in (2.2) to find zˆ i . Next, using (1.3), (1.4), (1.8) and (1.9) one can approximate the first two moments of the number in port i as well as the wait at port i. The symmetric case: If λi = λ, µiB = µB , and µiV = µV for all i then the analysis simplifies. From (2.2), φ=
λˆz − µV , µB zˆ − µV
(2.8)
and from (2.6) and (2.8), η=
µB zˆ + (n − 1)µV − nλˆz . n∆ˆz(µB − λ)
(2.9)
From (2.9), ηn∆(µB − λ)ˆz = (n − 1)µV + (µB − nλ)ˆz.
(2.10)
Also, from (A.4), zˆ solves the equation λˆz2 − ηˆz − (λ + µV )ˆz + µV = 0.
(2.11)
46
L.D. Servi, S.G. Finn / Performance Evaluation 50 (2002) 41–52
Multiplying (2.11) by n∆(µB − λ), substituting in (2.10) and then simplifying, we get a zˆ 2 + bˆz + c = 0, where a = nλ∆(µB − λ), b = −µB + nλ − (λ + µV )n∆(µB − λ), c = µV (n∆(µB − λ) − n + 1), whose solution is √ −b + b2 − 4ac zˆ = . 2a
(2.12)
Therefore, from (1.3), (1.4), (1.8) and (1.9) the expected number in the system and waiting time as well as the corresponding variance can be found. Note that if µV = 0 then, from (2.9), η = (µB − nλ)/n∆(µB − λ) which is independent of zˆ and, from (2.11), zˆ = λ/(λ + η). 3. An example Consider an Internet Protocol (IP) access network, where n access routers are connected via an optical network to the global IP network through a gateway router [2]. As illustrated in Fig. 2, access router i has pi ports that connect to an optical network infrastructure. The gateway router has a sufficient number of ports to the optical network, P , as well as to the backbone so that one can assume there is no output queuing at the gateway router. Each port has a tunable optical transmitter and receiver and can transmit data over a single wavelength. The reconfiguration problem consists of determining which access router ports should be connected to the gateway router ports using which wavelengths. A particular configuration is also referred to as a logical or electronic layer topology. We assume that the optical network infrastructure is equipped with at least P wavelengths, thus the feasibility of the electronic layer topologies depends only
Fig. 2. An optical access network.
L.D. Servi, S.G. Finn / Performance Evaluation 50 (2002) 41–52
47
upon the port restrictions rather than on wavelength or physical topology restrictions. If we assume that the wavelengths associated with the transceivers on the gateway router ports are fixed, then reconfiguring the electronic layer topology requires re-tuning the transmitter on one edge router and stopping transmission on another edge router port. The process of reconfiguring the electronic layer topology results in a period of time where the wavelength being re-tuned cannot be used. The disadvantage of reconfiguration is that while one or more wavelengths are reconfiguring from one router to another the capacity associated with these wavelengths is unavailable to either router and therefore is wasted. On the other hand, reconfiguring has the potential to dynamically match the capacity with the immediate needs of the router. One approach is to never reconfigure in an attempt to minimize wasted capacity. The other extreme is to reconfigure based on the complete state of the system. The former strategy is best only when the reconfiguration time (relative to the service time of the queues) is very long [2]. The latter strategy will be inefficient if it: (i) requires information which takes excessive bandwidth that would otherwise be devoted to traffic, (ii) requires excessive computation time, or (iii) is not sufficiently robust with respect to having delayed information. An intermediate and more robust strategy would reconfigure based only on whether an access router is empty. A previous study [2] examined a variation of an exhaustive schedule, i.e., a schedule which assigns all wavelengths to router i and, when it first becomes empty, reconfigure all wavelengths to router i + 1. Below we examine a generalization of this strategy which was first proposed and examined by simulation in [5] for a related reconfiguration access problem applied to satellites. Router i has wavelengths
Fig. 3. Expected wait for a symmetric cyclic service queue λ = 0.14, µV = 0.2(1 − u∗ ), µB = u∗ + 0.2(1 − u∗ ), n = 5, ∆ = 10.
48
L.D. Servi, S.G. Finn / Performance Evaluation 50 (2002) 41–52
permanently assigned to it giving it the capability of being serviced at a nominal average rate µiV . In addition, roving wavelengths, with the capacity to service at an additional average rate of u∗ are initially configured to router 1 giving it a total average service rate of µ1B = µ1V + u∗ . If router i is currently operating at a total average service rate of µiB = µiV + u∗ and becomes empty at a service completion, the roving wavelengths are reconfigured to router (i + 1) mod(n). This results in router i instantaneously reducing its average service rate to µiV and, only after a reconfiguration delay of ∆, router (i + 1) mod(n) ∗ increasing its average service rate by u∗ to µi+1 = µi+1 B V + u . This cycle continually repeats itself. Given a fixed number of wavelengths, it is natural to consider the optimize number assigned to be permanent. We model this by either assuming that having multiple wavelengths serving a router can be accurately approximated by a single server or, alternately, by assuming that the multiple wavelengths are engineered to indeed perform as if they are a single server. The latter assumption, called the continuous bandwidth mode, is discussed in [2]. We also introduce the previously discussed approximations associated with applying a working vacation model to a cyclic service system. To apply this model to a five router symmetric system, we renormalize time so that the total average service rate, u∗ + ni=1 µiB , equals 1. Having the token corresponds to having the additional service capacity of u∗ . Hence, u∗ is also the fraction of the total service rate capacity that is cyclically roving from queue to queue and the total system load is 5λ. For Fig. 3, we consider a system with a 70% load (so λ = 0.70/5 = 0.14) and a reconfiguration time of 10. The results are compared to a simulation using a modification of a simulator developed for [2]. For values of u∗ > 0.5 the results are unreliable. However, one sees that the analysis follows the shape of the simulated exact results for smaller values of u∗ . This tends to be the region of most practical interest. While this analysis represents a step forward in the approximation of this class of cyclic queue systems, clearly more analysis is needed to find even more accurate predictions. This example illustrates that the use of a small fraction of roving wavelengths for WDM access networks can improve performance over the alternative of having no reconfiguration or reconfiguring all wavelengths. The use of working vacation models provides a first step in quantifying such improvements.
Acknowledgements The authors thank J. Cooley, V. Mehta, A. Narula-Tam, and S. Stadler, all of MIT, Lincoln Laboratory, for valuable discussions related to this paper. In addition, they thank T. Ott (New Jersey Institute of Technology) for correcting a previous interpretation of Eq. (1.1).
Appendix A. Proof of theorem The derivation uses classical methods. Motivated by the state transition rate diagram in Fig. A.1, the balance equations are (λ + χ{j =0} µV + χ{j =0} η)pV (j ) = µV pV (j + 1) + χ{j =0} λpV (j − 1) + χ{j =0} µB pB (1), j = 0, . . . , ∞,
(A.1)
L.D. Servi, S.G. Finn / Performance Evaluation 50 (2002) 41–52
49
Fig. A.1. State transition rate diagram.
and (λ + χ{j =0} µB )pB (j ) = χ{j =0} ηpV (j ) + χ{j =0} λpB (j − 1) + χ{j =0} µB pB (j + 1), j = 0, . . . , ∞, where
χ{A} =
1
if A is true,
0
if A is false.
(A.2)
Multiplying (A.1) by zj and summing for j = 0 to ∞, we get (λ + µV + η)PV (z) − (µV + η)pV (0) = µV z−1 (PV (z) − pV (0)) + λzPV (z) + µB pB (1). Hence, µV pV (0) + ηpV (0) − µV z−1 pV (0) + µB pB (1) λ + µV + η − µV z−1 − λz −z(ηpV (0) + µB pB (1)) + µV (1 − z)pV (0) = , A(z)
PV (z) =
(A.3)
where A(z) = λz2 − (η + λ + µV )z + µV = λ(z − z∗ )(z − zˆ )
for |ˆz| ≥ 1.
(A.4)
Applying the quadratic formula leads to the second equation in (1.5). Note, also that from (A.4), µV = λz∗ zˆ .
(A.5)
50
L.D. Servi, S.G. Finn / Performance Evaluation 50 (2002) 41–52
Regularity demands that the numerator of (A.3) equals 0 for z = z∗ . Therefore ηpV (0) + µB pB (1) = µV (z∗ )−1 (1 − z∗ )pV (0) = ((z∗ )−1 − 1)µV pV (0).
(A.6)
From (A.3) and (A.6), PV (z) =
−z((z∗ )−1 − 1) + (1 − z) µV pV (0) A(z)
=
(z∗ )−1 (z∗ − z) µV pV (0) λ(z − z∗ )(z − zˆ )
=
pV (0) 1 − zˆ −1 z
from (A.4)
using (A.5).
(A.7)
Eq. (A.7) is a startlingly simple result: it implies that the number in the system conditioned on being in a working vacation is geometrically distributed. Intuitively, this is a consequence of having a constant hazard rate of leaving each state, λ + η, and a constant hazard rate of entering each state, µV , except at j = 0. The analysis of the states corresponding to not being in a working vacation is similar: multiplying (A.2) by zj and summing for j = 0 to ∞, using the observation that (A.2) implies that pB (0) = 0, we get (λ + µB )PB (z) = η(PV (z) − pV (0)) + λzPB (z) + µB z−1 (PB (z) − pB (1)z). Hence, PB (z) =
ηPV (z) − (ηpV (0) + µB pB (1)) . λ + µB − µB z−1 − λz
(A.8)
From (A.6) and (A.8), PB (z) =
ηPV (z) − µV ((z∗ )−1 − 1)pV (0) λ + µB − µB z−1 − λz
=
η/(1 − zˆ −1 z) − λˆz + µV pV (0) (1 − z)(λ − µB z−1 )
=
−ˆz−1 µV z + λz − λˆz + µV + η pV (0) −(1 − z)µB z−1 (1 − (λ/µB )z)(1 − zˆ −1 z)
from (A.5) and (A.7)
−ˆz−1 µV z + λz − λ + µV zˆ −1 pV (0) from (A.4) since (ˆz)−1 A(ˆz) = 0 −(1 − z)µB z−1 (1 − (λ/µB )z)(1 − zˆ −1 z) λ − µV zˆ −1 z pV (0) (A.9) = (1 − (λ/µB )z)(1 − zˆ −1 z) µB 1 1 λ − µV zˆ −1 = pV (0). (A.10) − 1 − (λ/µB )z 1 − zˆ −1 z λ − µB zˆ −1
=
L.D. Servi, S.G. Finn / Performance Evaluation 50 (2002) 41–52
51
From (A.7) and (A.9), [((λ − µV zˆ −1 )/µB )z] + (1 − (λ/µB )z) pV (0) (1 − (λ/µB )z)(1 − zˆ −1 z) 1 − (µV /µB zˆ )z = pV (0). (1 − (λ/µB )z)(1 − zˆ −1 z)
P (z) = PB (z) + PV (z) =
This and P (1) = 1 imply pV (0) =
(1 − (λ/µB ))(1 − zˆ −1 ) , 1 − (µV /µB zˆ )
(A.11)
the first equation in (1.1) and (1.3) and Eq. (1.4) since E[N] = P (1) and Var[N] = P (1) + P (1) − (P (1))2 . From (A.7) and (A.10) 1 1 λ − µV zˆ −1 λ − µV zˆ −1 P (z) = p pV (0) (0) + 1 − V 1 − (λ/µB )z λ − µB zˆ −1 1 − zˆ −1 z λ − µB zˆ −1 (1 − zˆ −1 )(λ − µV zˆ −1 ) (1 − λ/µB )(µV − µB )ˆz−1 1 − λ/µB 1 − zˆ −1 = + , 1 − (λ/µB )z (1 − µV /µB zˆ )(λ − µB zˆ −1 ) 1 − zˆ −1 z (1 − (µV /µB zˆ ))(λ − µB zˆ −1 ) which implies the second equality in (1.1) and (1.3). A.1. Derivation of the coefficients of (2.8) From (2.2), φ i (µiB zˆ i − µiV ) = λi zˆ i − µiV so zˆ i (φ i µiB − λi ) = −µiV (1 − φ i ).
(A.12)
But from (A.4), (φ i µiB − λi )2 (λi (ˆzi )2 − ηi zˆ i − (λi + µiV )ˆzi + µiV ) = 0.
(A.13)
Combining (A.12), (A.13) and (2.6), 1 − nj=1 φ j (φ i µiB − λi )µiV λi (µiV )2 (1 − φ i )2 + + (λi + µiV )µiV (1 − φ i )(φ i µiB − λi ) n∆ + µiV (φ i µiB − λi )2 = 0, (A.14) which simplifies to (2.7) where ai = λi (µiV )2 − (λi + µiV )µiB µiV + µiV (µiB )2 = (λi − µiB )(µiV − µiB )µiV , bi = −2λi (µiV )2 + =
(A.15)
µiB µiV + (λi + µiV )µiV (µiB + λi ) − 2µiB µiV λi n∆
µiB µiV + µiV (λi − µiV )(λi − µiB ), n∆
(A.16)
52
L.D. Servi, S.G. Finn / Performance Evaluation 50 (2002) 41–52
ci = −
µiB µiV , n∆
di = λi (µiV )2 −
(A.17) λi µiV λi µiV − (λi + µiV )λi µiV + µiV (λi )2 = − , n∆ n∆
(A.18)
and ei =
λi µiV . n∆
(A.19)
References [1] J. Keilson, L.D. Servi, A distributional form of Little’s law, Oper. Res. Lett. 7 (5) (1988) 223–227. [2] A. Narula-Tam, S.G. Finn, M. Médard, Analysis of reconfiguration in IP over WDM access networks, in: Proceedings of the Optical Fiber Communication Conference (OFC), 2001, pp. MN4.1–MN4.3. [3] L.D. Servi, Average delay approximation of M/G/1 cyclic service queues with Bernoulli schedules, IEEE J. Select. Areas Commun. SAC-4 (6) (1986) 813–822. [4] H. Takagi, Queueing analysis of polling models: an update, in: H. Takagi (Ed.), Stochastic Analysis of Computer and Communication Systems, North-Holland, Amsterdam, 1990, pp. 267–318. [5] C.-H. Lee, Integrated dynamic bandwidth allocation and congestion control in satellite frame relay networks, M.S. Thesis, Department of EECS, Massachusetts Institute of Technology, Lexington, MA, 1994. L.D. Servi was born in Buffalo, NY in 1955. He received a M.S. and Ph.D. in engineering from Harvard University in 1978 and 1981, respectively, and a joint Sc.B. - Sc.M. degree in Applied Mathematics from Brown University in 1977. Dr. Servi joined the technical staff of MIT Lincoln in 2000. Previously, he worked at GTE Laboratories on both theoretical and applied telecommunication projects. In 1999, he was a joint recipient GTE’s highest technical honor, given by CEO Charles R. Lee, for the development and implementation of an inventory policy related to new telephone lines which had a substantial financial impact. Previously, he worked at Bell Laboratories for two years. In 1989 he took a nine month sabbatical leave and split his time between M.I.T.’s Department of Electrical Engineering and Computer Science and Harvard University’s Division of Applied Sciences. Dr. Servi is a former editor of Operations Research and OR Journal on Computing, a former Chair of the INFORMS Technical Section on Telecommunications, the current Chair of the INFORMS Applied Probability Society, a member of the INFORMS Board of Directors since 1998, and a member of the Executive Committee of the INFORMS Board of Directors since 2001.
Steven Finn was born in Boston, MA in 1946. He received B.S., M.S. and ScD. electrical engineering degrees from MIT in 1969 and 1975, respectively. From 1975 to 1980 he worked for Codex/Motorola Corporation where he held various R&D positions including Director of Network Product Development and Director of Network Research. While at Codex he was also a member of ANSI and CCITT (ITU-T) committees involved in networking standards development. In 1980 he founded Bytex Corporation, a data communications equipment manufacturer. Dr. Finn held the position of CEO and Chairman of the Board through 1987 and 1990, respectively. In 1990 Dr. Finn returned to MIT as a Vinton Hayes Fellow and Visiting Scientist in the Laboratory of Information and Decision Sciences. Currently he is a Principal Research Scientist at MIT and a Sr. Member of the Technical Staff at the Lincoln Laboratory. Dr. Finn teaches and supervises graduate thesis research in the Department of Electrical Engineering and Computer Science. His current research interests are in the areas of optical networks, high-speed data network transport, network architecture and network management. In addition to his work at MIT, Dr. Finn is a special limited partner and consultant to Matrix Partners a leading Boston venture capital firm. He is also an advisor and member of the Board of Directors of several high technology start-up companies.
M/M/1 queues with working vacations (M/M/1/WV)夽 L.D. Servi∗ , S.G. Finn Lincoln Laboratory, Massachusetts Institute of Technology, 244 Wood Street, Lexington, MA 02420-9180, USA Received 5 June 2001; received in revised form 25 January 2002
Abstract The classical single server vacation model is generalized to consider a server which works at a different rate rather than completely stops during the vacation period. Simple explicit formulae for the mean, variance, and distribution of the number and time in the system are presented. The distributional results generalize the classical vacation decomposition with no service during a vacation. This model approximates a multi-queue system whose service rate is one of the two speeds for which the fast speed mode cyclically moves from queue to queue with an exhaustive schedule. This work is motivated and illustrated by the analysis of a WDM optical access network using multiple wavelengths which can be reconfigured. © 2002 Elsevier Science B.V. All rights reserved. Keywords: M/M/1 queues; Vacation model; Working vacation; WDM optical access
1. A single queue working vacation system Consider a single server queue which begins a working vacation of mean duration 1/η when the system is empty. If the system is empty when a working vacation ends, another begins. During the working vacation customers are served at a mean rate of µV . When the server is not on a working vacation customers are served at a mean rate of µB . The interarrival times, service times and vacation times are all exponentially distributed. Let N be the number in the system, λ the arrival rate, 1/η the average duration of a working vacation, 1/µB the average service time when the server is not on a working vacation, 1/µV the average service time when the server is on a working vacation, pB (j ) the probability of j in the system when the server is not on a working vacation, pj V (j ) the probability of j in the system when the server is on working vacation, Pi (z) = ∞ j =0 pi (j )z for i = B or V , and P (z) = PB (z) + PV (z). The following theorem uses classical methods to prove that N plus a geometrical distributed random variable equals the number in the system in the absence of vacations plus another geometrically distributed 夽 This work is sponsored by DARPA under contract F19628-00-C-0002. Opinions, interpretations, recommendations and conclusions are those of the authors and are not necessarily endorsed by the Department of Defense. ∗ Corresponding author. E-mail addresses: [email protected] (L.D. Servi), [email protected] (S.G. Finn).
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 0 5 7 - 3
42
L.D. Servi, S.G. Finn / Performance Evaluation 50 (2002) 41–52
random variable, or, equivalently, N is the weighted average of two geometrical distributions. If µV = 0 these results specialize to the known decomposition for M/M/1 systems with exponential vacations. Theorem. The number in the system for an M/M/1 with exponential working vacation has an z transform 1 − λ/µB 1 − zˆ −1 1 − µV /ˆzµB −1 P (z) = 1 − (λ/µB )z 1 − zˆ −1 z 1 − (µV /ˆzµB )z 1 − λ/µB 1 − zˆ −1 =γ + (1 − γ ) , (1.1) 1 − (λ/µB )z 1 − zˆ −1 z a distribution
Pr[N = j ] = γ
1−
λ µB
λ µB
j
+ (1 − γ )(1 − zˆ −1 )(ˆz)−j ,
(1.2)
a mean E[N] =
λ 1 µV λ 1 + − =γ + (1 − γ ) , µB − λ zˆ − 1 zˆ µB − µV µB − λ zˆ − 1
(1.3)
and a variance Var[N] =
zˆ µV µB zˆ λµB + − , 2 2 (µB − λ) (ˆz − 1) (µB zˆ − µV )2
where (ˆz − 1)(λˆz − µV )µB γ = (µB zˆ − µV )(λˆz − µB )
and
zˆ =
λ + µV + η +
(1.4)
(λ + µV + η)2 − 4λµV . 2λ
(1.5)
Proof. See Appendix A.
Three cases are considered which specialize to previously known results: (i) If the system never takes a vacation, i.e., µB = µV or if η (and hence zˆ ) approaches ∞, from (1.5), γ = 1 so, from (1.1) N is a geometrical distribution with a rate λ/µB ; (ii) if the system is always on a working vacation, i.e., η = 0, then from (1.5), zˆ = µV /λ and γ = 0. Hence, from (1.1), N is a geometrical distribution with a rate 1/ˆz = λ/µV ; (iii) if the server does no work during a working vacation, i.e., µV = 0 then from (1.5), zˆ = (λ + η)/λ. Hence, from the first equation in (1.1), 1 − λ/µB η P (z) = , 1 − (λ/µB )z η + λ − zλ which is the classical vacation decomposition. From (A.7), Pr[N = 0|i = V ] = 1 −
Pr[N = 0 and i = V ] pV (0) =1− = zˆ −1 , Pr[i = V ] PV (1)
(1.6)
where i ∈ {B, V } is the state variable denoting whether the system is in a regular busy period or in a working vacation. Eq. (1.6) provides an interpretation of zˆ as well as is useful in Section 2.
L.D. Servi, S.G. Finn / Performance Evaluation 50 (2002) 41–52
43
Fig. 1. Distribution for the number in system for a working vacation model λ = 2, µB = 3.5, µV = 1.5, η = 0.05 and j = 0, . . . , 50.
The second equality in (1.1) is useful to compute (1.2). However, note that none of the analysis guarantees that 0 ≤ γ ≤ 1 (even if the system is stable). For example, if λ = 2, µB = 3.5, µV = 1.5, and η = 0.05 then γ ≈ −0.06, but as illustrated in Fig. 1, N has a well-behaved distribution with a mean of approximately 12.9. Corollary. The total time in the system for an M/M/1 with exponential working vacations has a Laplace transform −1 µB − λ λˆz − λ λˆzµB /µV − λ αw (s) = , (1.7) s + µB − λ s + λˆz − λ s + λˆzµB /µV − λ a mean E[W ] =
1 µV 1 + − , µB − λ λ(ˆz − 1) λ(ˆzµB − µV )
(1.8)
and a variance Var[W ] =
1 µ2V 1 + − . (µB − λ)2 λ2 (ˆz − 1)2 λ2 (µB zˆ − µV )2
(1.9)
Proof. If αw (s) is the Laplace transform of the time in the system, then Keilson and Servi [1] states four
44
L.D. Servi, S.G. Finn / Performance Evaluation 50 (2002) 41–52
necessary conditions for P (z) = αw (λ − λz).
(1.10)
(a) The arrivals are Poisson; (b) all arriving customers enter the system, and remain in the system until served; (c) the customers leave the system one at a time in the order of arrivals; and (d) for any time t, the arrival process after time t and the time in system of any customer arriving before t are independent. Since these four conditions clearly hold, (1.10) and therefore, using (1.1), (1.7) follows. Taking the derivative of (1.10) and evaluating at z = 0 leads to the classical Little’s law formula, E[W ] = E[N]/λ, and hence, from (1.3), (1.8) follows. Taking a second derivative of (1.10) and evaluating at z = 0, leads to Var[W ] = (Var[N] − E[N ])/λ2 so, using (1.2) and (1.3), (1.9) follows. 2. Application to multiple queue systems Consider the following n-queue generalization of a cyclic service queue model which is motivated by a reconfigurable WDM optical access network: a single token cyclically visits each queue. Each queue operates either at a fast service rate or a nominal service rate. A queue operates at its fast service rate as soon as it acquires the token. If queue i possesses the token (and hence is operating at its fast speed) and becomes empty, its service rate is instantly reduced to its nominal rate for future arrivals but the token requires a reconfiguration time to pass before it visits queue (i + 1) mod(n). This cycle continually repeats itself. Following previous cyclic service queue analysis, e.g., [3,4], one notes that from the point of view of queue i the server follows a working vacation model. Hence, the previous analysis can be applied if the working vacation time corresponding to queue i can be determined. However, to do so implicitly introduces the following two approximating assumptions: (i) the duration of the working vacation of queue i is exponentially distributed, and (ii) the duration of the working vacation is an independent random variable uncorrelated with the queue i’s busy period. These approximations were previously used for a related classical cyclic service queues (e.g., [3]) with some success although under very high load the impact of ignoring correlations can be problematic. To proceed, let λi be the arrival rate to queue i, µiB be the fast service rate of queue i, i.e., the rate when it possesses the token, µiV the nominal service rate at queue i, V i the average duration that queue i is operating with its nominal service rate mode, i.e., the working vacation duration, V i = 1/ηi , zˆ i the probability of queue i not empty given the token is not at queue i, cf. (1.5) and (1.6), C the average cycle time, i.e., the time between token visits to queue 1, B i the average busy period, i.e., the average time the token is at queue i. Note that on a given cycle this could equal zero. And ∆ be the average reconfiguration time, i.e., the time needed to switch from a nominal rate to a fast rate (it is assumed that switching from a fast to a nominal rate is instantaneous). If the system is stable, the average number of arrivals to queue i in a cycle must equal the average number of service completions during a cycle. Hence, for i = 0, . . . , n − 1, λi C = µiB B i + µiV (C − B i ) Pr[not idle|in a working vacation period]. From (1.6), this equals µiB B i + µiV (C − B i )/ˆzi so B i = φ i C,
(2.1)
i
where φ is defined by φi =
λi zˆ i − µiV . µiB zˆ i − µiV
(2.2)
L.D. Servi, S.G. Finn / Performance Evaluation 50 (2002) 41–52
45
The average cycle time, C, equals the total reconfiguration time plus the total time spent in the fast service time speed at each queue, i.e., C = n∆ +
n
Bj .
(2.3)
j =1
Combining (2.1) and (2.3) leads to C=
1−
n∆ n
j =1
φj
.
(2.4)
Finally, the average working vacation for queue i is set equal to the average time it is not working at the fast service time speed, i.e., 1 = V i = C − Bi. ηi
(2.5)
From (2.1), (2.4) and (2.5), 1 − nj=1 φ j i η = . n∆(1 − φ i )
(2.6)
As derived in Appendix A, the φ i ’s are a solution to ai (φ i )2 + bi φ i + ci φ i φˆ + di + ei φˆ = 0,
i = 1, . . . , n,
(2.7) n
where ai , bi , ci , di , and ei are given in (A.15)–(A.19) and φˆ = j =1 φ j . Eq. (2.7) can be solved by ˆ solving the n, now independent, quadratic equations in terms iteratively selecting an initial value of φ, ˆ The solution of φˆ and then φ i , i = 1, . . . , n of φˆ and then finding and selecting an improved value of φ. can be used in (2.2) to find zˆ i . Next, using (1.3), (1.4), (1.8) and (1.9) one can approximate the first two moments of the number in port i as well as the wait at port i. The symmetric case: If λi = λ, µiB = µB , and µiV = µV for all i then the analysis simplifies. From (2.2), φ=
λˆz − µV , µB zˆ − µV
(2.8)
and from (2.6) and (2.8), η=
µB zˆ + (n − 1)µV − nλˆz . n∆ˆz(µB − λ)
(2.9)
From (2.9), ηn∆(µB − λ)ˆz = (n − 1)µV + (µB − nλ)ˆz.
(2.10)
Also, from (A.4), zˆ solves the equation λˆz2 − ηˆz − (λ + µV )ˆz + µV = 0.
(2.11)
46
L.D. Servi, S.G. Finn / Performance Evaluation 50 (2002) 41–52
Multiplying (2.11) by n∆(µB − λ), substituting in (2.10) and then simplifying, we get a zˆ 2 + bˆz + c = 0, where a = nλ∆(µB − λ), b = −µB + nλ − (λ + µV )n∆(µB − λ), c = µV (n∆(µB − λ) − n + 1), whose solution is √ −b + b2 − 4ac zˆ = . 2a
(2.12)
Therefore, from (1.3), (1.4), (1.8) and (1.9) the expected number in the system and waiting time as well as the corresponding variance can be found. Note that if µV = 0 then, from (2.9), η = (µB − nλ)/n∆(µB − λ) which is independent of zˆ and, from (2.11), zˆ = λ/(λ + η). 3. An example Consider an Internet Protocol (IP) access network, where n access routers are connected via an optical network to the global IP network through a gateway router [2]. As illustrated in Fig. 2, access router i has pi ports that connect to an optical network infrastructure. The gateway router has a sufficient number of ports to the optical network, P , as well as to the backbone so that one can assume there is no output queuing at the gateway router. Each port has a tunable optical transmitter and receiver and can transmit data over a single wavelength. The reconfiguration problem consists of determining which access router ports should be connected to the gateway router ports using which wavelengths. A particular configuration is also referred to as a logical or electronic layer topology. We assume that the optical network infrastructure is equipped with at least P wavelengths, thus the feasibility of the electronic layer topologies depends only
Fig. 2. An optical access network.
L.D. Servi, S.G. Finn / Performance Evaluation 50 (2002) 41–52
47
upon the port restrictions rather than on wavelength or physical topology restrictions. If we assume that the wavelengths associated with the transceivers on the gateway router ports are fixed, then reconfiguring the electronic layer topology requires re-tuning the transmitter on one edge router and stopping transmission on another edge router port. The process of reconfiguring the electronic layer topology results in a period of time where the wavelength being re-tuned cannot be used. The disadvantage of reconfiguration is that while one or more wavelengths are reconfiguring from one router to another the capacity associated with these wavelengths is unavailable to either router and therefore is wasted. On the other hand, reconfiguring has the potential to dynamically match the capacity with the immediate needs of the router. One approach is to never reconfigure in an attempt to minimize wasted capacity. The other extreme is to reconfigure based on the complete state of the system. The former strategy is best only when the reconfiguration time (relative to the service time of the queues) is very long [2]. The latter strategy will be inefficient if it: (i) requires information which takes excessive bandwidth that would otherwise be devoted to traffic, (ii) requires excessive computation time, or (iii) is not sufficiently robust with respect to having delayed information. An intermediate and more robust strategy would reconfigure based only on whether an access router is empty. A previous study [2] examined a variation of an exhaustive schedule, i.e., a schedule which assigns all wavelengths to router i and, when it first becomes empty, reconfigure all wavelengths to router i + 1. Below we examine a generalization of this strategy which was first proposed and examined by simulation in [5] for a related reconfiguration access problem applied to satellites. Router i has wavelengths
Fig. 3. Expected wait for a symmetric cyclic service queue λ = 0.14, µV = 0.2(1 − u∗ ), µB = u∗ + 0.2(1 − u∗ ), n = 5, ∆ = 10.
48
L.D. Servi, S.G. Finn / Performance Evaluation 50 (2002) 41–52
permanently assigned to it giving it the capability of being serviced at a nominal average rate µiV . In addition, roving wavelengths, with the capacity to service at an additional average rate of u∗ are initially configured to router 1 giving it a total average service rate of µ1B = µ1V + u∗ . If router i is currently operating at a total average service rate of µiB = µiV + u∗ and becomes empty at a service completion, the roving wavelengths are reconfigured to router (i + 1) mod(n). This results in router i instantaneously reducing its average service rate to µiV and, only after a reconfiguration delay of ∆, router (i + 1) mod(n) ∗ increasing its average service rate by u∗ to µi+1 = µi+1 B V + u . This cycle continually repeats itself. Given a fixed number of wavelengths, it is natural to consider the optimize number assigned to be permanent. We model this by either assuming that having multiple wavelengths serving a router can be accurately approximated by a single server or, alternately, by assuming that the multiple wavelengths are engineered to indeed perform as if they are a single server. The latter assumption, called the continuous bandwidth mode, is discussed in [2]. We also introduce the previously discussed approximations associated with applying a working vacation model to a cyclic service system. To apply this model to a five router symmetric system, we renormalize time so that the total average service rate, u∗ + ni=1 µiB , equals 1. Having the token corresponds to having the additional service capacity of u∗ . Hence, u∗ is also the fraction of the total service rate capacity that is cyclically roving from queue to queue and the total system load is 5λ. For Fig. 3, we consider a system with a 70% load (so λ = 0.70/5 = 0.14) and a reconfiguration time of 10. The results are compared to a simulation using a modification of a simulator developed for [2]. For values of u∗ > 0.5 the results are unreliable. However, one sees that the analysis follows the shape of the simulated exact results for smaller values of u∗ . This tends to be the region of most practical interest. While this analysis represents a step forward in the approximation of this class of cyclic queue systems, clearly more analysis is needed to find even more accurate predictions. This example illustrates that the use of a small fraction of roving wavelengths for WDM access networks can improve performance over the alternative of having no reconfiguration or reconfiguring all wavelengths. The use of working vacation models provides a first step in quantifying such improvements.
Acknowledgements The authors thank J. Cooley, V. Mehta, A. Narula-Tam, and S. Stadler, all of MIT, Lincoln Laboratory, for valuable discussions related to this paper. In addition, they thank T. Ott (New Jersey Institute of Technology) for correcting a previous interpretation of Eq. (1.1).
Appendix A. Proof of theorem The derivation uses classical methods. Motivated by the state transition rate diagram in Fig. A.1, the balance equations are (λ + χ{j =0} µV + χ{j =0} η)pV (j ) = µV pV (j + 1) + χ{j =0} λpV (j − 1) + χ{j =0} µB pB (1), j = 0, . . . , ∞,
(A.1)
L.D. Servi, S.G. Finn / Performance Evaluation 50 (2002) 41–52
49
Fig. A.1. State transition rate diagram.
and (λ + χ{j =0} µB )pB (j ) = χ{j =0} ηpV (j ) + χ{j =0} λpB (j − 1) + χ{j =0} µB pB (j + 1), j = 0, . . . , ∞, where
χ{A} =
1
if A is true,
0
if A is false.
(A.2)
Multiplying (A.1) by zj and summing for j = 0 to ∞, we get (λ + µV + η)PV (z) − (µV + η)pV (0) = µV z−1 (PV (z) − pV (0)) + λzPV (z) + µB pB (1). Hence, µV pV (0) + ηpV (0) − µV z−1 pV (0) + µB pB (1) λ + µV + η − µV z−1 − λz −z(ηpV (0) + µB pB (1)) + µV (1 − z)pV (0) = , A(z)
PV (z) =
(A.3)
where A(z) = λz2 − (η + λ + µV )z + µV = λ(z − z∗ )(z − zˆ )
for |ˆz| ≥ 1.
(A.4)
Applying the quadratic formula leads to the second equation in (1.5). Note, also that from (A.4), µV = λz∗ zˆ .
(A.5)
50
L.D. Servi, S.G. Finn / Performance Evaluation 50 (2002) 41–52
Regularity demands that the numerator of (A.3) equals 0 for z = z∗ . Therefore ηpV (0) + µB pB (1) = µV (z∗ )−1 (1 − z∗ )pV (0) = ((z∗ )−1 − 1)µV pV (0).
(A.6)
From (A.3) and (A.6), PV (z) =
−z((z∗ )−1 − 1) + (1 − z) µV pV (0) A(z)
=
(z∗ )−1 (z∗ − z) µV pV (0) λ(z − z∗ )(z − zˆ )
=
pV (0) 1 − zˆ −1 z
from (A.4)
using (A.5).
(A.7)
Eq. (A.7) is a startlingly simple result: it implies that the number in the system conditioned on being in a working vacation is geometrically distributed. Intuitively, this is a consequence of having a constant hazard rate of leaving each state, λ + η, and a constant hazard rate of entering each state, µV , except at j = 0. The analysis of the states corresponding to not being in a working vacation is similar: multiplying (A.2) by zj and summing for j = 0 to ∞, using the observation that (A.2) implies that pB (0) = 0, we get (λ + µB )PB (z) = η(PV (z) − pV (0)) + λzPB (z) + µB z−1 (PB (z) − pB (1)z). Hence, PB (z) =
ηPV (z) − (ηpV (0) + µB pB (1)) . λ + µB − µB z−1 − λz
(A.8)
From (A.6) and (A.8), PB (z) =
ηPV (z) − µV ((z∗ )−1 − 1)pV (0) λ + µB − µB z−1 − λz
=
η/(1 − zˆ −1 z) − λˆz + µV pV (0) (1 − z)(λ − µB z−1 )
=
−ˆz−1 µV z + λz − λˆz + µV + η pV (0) −(1 − z)µB z−1 (1 − (λ/µB )z)(1 − zˆ −1 z)
from (A.5) and (A.7)
−ˆz−1 µV z + λz − λ + µV zˆ −1 pV (0) from (A.4) since (ˆz)−1 A(ˆz) = 0 −(1 − z)µB z−1 (1 − (λ/µB )z)(1 − zˆ −1 z) λ − µV zˆ −1 z pV (0) (A.9) = (1 − (λ/µB )z)(1 − zˆ −1 z) µB 1 1 λ − µV zˆ −1 = pV (0). (A.10) − 1 − (λ/µB )z 1 − zˆ −1 z λ − µB zˆ −1
=
L.D. Servi, S.G. Finn / Performance Evaluation 50 (2002) 41–52
51
From (A.7) and (A.9), [((λ − µV zˆ −1 )/µB )z] + (1 − (λ/µB )z) pV (0) (1 − (λ/µB )z)(1 − zˆ −1 z) 1 − (µV /µB zˆ )z = pV (0). (1 − (λ/µB )z)(1 − zˆ −1 z)
P (z) = PB (z) + PV (z) =
This and P (1) = 1 imply pV (0) =
(1 − (λ/µB ))(1 − zˆ −1 ) , 1 − (µV /µB zˆ )
(A.11)
the first equation in (1.1) and (1.3) and Eq. (1.4) since E[N] = P (1) and Var[N] = P (1) + P (1) − (P (1))2 . From (A.7) and (A.10) 1 1 λ − µV zˆ −1 λ − µV zˆ −1 P (z) = p pV (0) (0) + 1 − V 1 − (λ/µB )z λ − µB zˆ −1 1 − zˆ −1 z λ − µB zˆ −1 (1 − zˆ −1 )(λ − µV zˆ −1 ) (1 − λ/µB )(µV − µB )ˆz−1 1 − λ/µB 1 − zˆ −1 = + , 1 − (λ/µB )z (1 − µV /µB zˆ )(λ − µB zˆ −1 ) 1 − zˆ −1 z (1 − (µV /µB zˆ ))(λ − µB zˆ −1 ) which implies the second equality in (1.1) and (1.3). A.1. Derivation of the coefficients of (2.8) From (2.2), φ i (µiB zˆ i − µiV ) = λi zˆ i − µiV so zˆ i (φ i µiB − λi ) = −µiV (1 − φ i ).
(A.12)
But from (A.4), (φ i µiB − λi )2 (λi (ˆzi )2 − ηi zˆ i − (λi + µiV )ˆzi + µiV ) = 0.
(A.13)
Combining (A.12), (A.13) and (2.6), 1 − nj=1 φ j (φ i µiB − λi )µiV λi (µiV )2 (1 − φ i )2 + + (λi + µiV )µiV (1 − φ i )(φ i µiB − λi ) n∆ + µiV (φ i µiB − λi )2 = 0, (A.14) which simplifies to (2.7) where ai = λi (µiV )2 − (λi + µiV )µiB µiV + µiV (µiB )2 = (λi − µiB )(µiV − µiB )µiV , bi = −2λi (µiV )2 + =
(A.15)
µiB µiV + (λi + µiV )µiV (µiB + λi ) − 2µiB µiV λi n∆
µiB µiV + µiV (λi − µiV )(λi − µiB ), n∆
(A.16)
52
L.D. Servi, S.G. Finn / Performance Evaluation 50 (2002) 41–52
ci = −
µiB µiV , n∆
di = λi (µiV )2 −
(A.17) λi µiV λi µiV − (λi + µiV )λi µiV + µiV (λi )2 = − , n∆ n∆
(A.18)
and ei =
λi µiV . n∆
(A.19)
References [1] J. Keilson, L.D. Servi, A distributional form of Little’s law, Oper. Res. Lett. 7 (5) (1988) 223–227. [2] A. Narula-Tam, S.G. Finn, M. Médard, Analysis of reconfiguration in IP over WDM access networks, in: Proceedings of the Optical Fiber Communication Conference (OFC), 2001, pp. MN4.1–MN4.3. [3] L.D. Servi, Average delay approximation of M/G/1 cyclic service queues with Bernoulli schedules, IEEE J. Select. Areas Commun. SAC-4 (6) (1986) 813–822. [4] H. Takagi, Queueing analysis of polling models: an update, in: H. Takagi (Ed.), Stochastic Analysis of Computer and Communication Systems, North-Holland, Amsterdam, 1990, pp. 267–318. [5] C.-H. Lee, Integrated dynamic bandwidth allocation and congestion control in satellite frame relay networks, M.S. Thesis, Department of EECS, Massachusetts Institute of Technology, Lexington, MA, 1994. L.D. Servi was born in Buffalo, NY in 1955. He received a M.S. and Ph.D. in engineering from Harvard University in 1978 and 1981, respectively, and a joint Sc.B. - Sc.M. degree in Applied Mathematics from Brown University in 1977. Dr. Servi joined the technical staff of MIT Lincoln in 2000. Previously, he worked at GTE Laboratories on both theoretical and applied telecommunication projects. In 1999, he was a joint recipient GTE’s highest technical honor, given by CEO Charles R. Lee, for the development and implementation of an inventory policy related to new telephone lines which had a substantial financial impact. Previously, he worked at Bell Laboratories for two years. In 1989 he took a nine month sabbatical leave and split his time between M.I.T.’s Department of Electrical Engineering and Computer Science and Harvard University’s Division of Applied Sciences. Dr. Servi is a former editor of Operations Research and OR Journal on Computing, a former Chair of the INFORMS Technical Section on Telecommunications, the current Chair of the INFORMS Applied Probability Society, a member of the INFORMS Board of Directors since 1998, and a member of the Executive Committee of the INFORMS Board of Directors since 2001.
Steven Finn was born in Boston, MA in 1946. He received B.S., M.S. and ScD. electrical engineering degrees from MIT in 1969 and 1975, respectively. From 1975 to 1980 he worked for Codex/Motorola Corporation where he held various R&D positions including Director of Network Product Development and Director of Network Research. While at Codex he was also a member of ANSI and CCITT (ITU-T) committees involved in networking standards development. In 1980 he founded Bytex Corporation, a data communications equipment manufacturer. Dr. Finn held the position of CEO and Chairman of the Board through 1987 and 1990, respectively. In 1990 Dr. Finn returned to MIT as a Vinton Hayes Fellow and Visiting Scientist in the Laboratory of Information and Decision Sciences. Currently he is a Principal Research Scientist at MIT and a Sr. Member of the Technical Staff at the Lincoln Laboratory. Dr. Finn teaches and supervises graduate thesis research in the Department of Electrical Engineering and Computer Science. His current research interests are in the areas of optical networks, high-speed data network transport, network architecture and network management. In addition to his work at MIT, Dr. Finn is a special limited partner and consultant to Matrix Partners a leading Boston venture capital firm. He is also an advisor and member of the Board of Directors of several high technology start-up companies.