2 light traffic derivatives for the GI/G/ queue. We shall see that our proof of analyticity is mainly based on some recursive formulas very similar to

Transcription

1 Analyticity of Single-Server Queues in Light Traffic Jian-Qiang Hu Manufacturing Engineering Department Boston University Cummington Street Boston, MA 0225 September 993; Revised May 99 Abstract Recently, several methods have been proposed to approximate performance measures of queueing systems based on their light traffic derivatives, e.g., the MacLaurin expansion, the Padé approximation, and interpolation with heavy traffic limits. The key condition required in all these approximations is that the performance measures be analytic when the arrival rates equal to zero. In this paper, we study the GI/G/ queue. We show that if the c.d.f. of the interarrival time can be expressed as a MacLaurin series over [0;), then the mean steady-state system time of a job is indeed analytic when the arrival rate to the queue equals to zero. This condition is satisfied by phase-type distributions but not c.d.f.'s without support [0; ), such as uniform and shifted exponential distributions. In fact, we show through two examples that the analyticity does not hold for most commonly used distribution functions which do not satisfy this condition. Keywords: The GI/G/ queue, light traffic derivatives, MacLaurin series, analyticity Short Title: Analyticity of the GI/GI/ queue Introduction In this paper, we study the analyticity of the mean (in fact, all the moments of) steadystate system time of a job in the GI/G/ queue in light traffic, i.e., when the arrival rate to the queue, denoted by, approaches zero. Our work is mainly motivated by several approximation methods recently developed which use the light traffic derivatives of the mean system time (i.e., the derivatives with respect to at 0) to approximate the entire response curve of the mean system time with respect to. In what follows, we first briefly review previous works on the subject. Benes [2] was among the first to study light traffic derivatives for queueing systems. Since then, many papers have been written on the subject of light traffic derivatives, e.g., see Reiman and Siman [3], Daley and Rolski [5], Asmussen [], Gong and Hu [7], Sigman [], and refereces therein. In particular, Gong and Hu's work is closely related to our work in this paper. In Gong and Hu [7], a simple recursive procedure is derived to calculate the

2 2 light traffic derivatives for the GI/G/ queue. We shall see that our proof of analyticity is mainly based on some recursive formulas very similar to those derived there. An immediate application of the light traffic derivatives is to use them to approximate the mean system time via a MacLaurin series. However, the MacLaurin series may result in poor approximations due to its slow convergence rate, especially for heavy traffic cases. Furthermore, as noted in Gong and Hu [7], for many GI/G/ queues, including some wellbehaved queues such as the E 2 /M/ queue with exponential service times and two-stage Erlang interarrival times, the MacLaurin series only converges in a small" neighborhood of 0. One way toovercome this problem to use the light traffic derivatives in conjunction with heavy traffic limits, e.g., see Burman and Smith [3, ], Reiman and Simon [2], and Fendick and Whitt [6]. (In fact, these authors approximate the mean system time by ratios of polynomial functions and a normalization factor ρ, where ρ is the traffic intensity.) Another method recently proposed by Gong, Nananukul, and Yan [8] is to use the Padé approximation method to approximate the mean system time based on the light traffic derivatives. Basically, inthepadé approximation, we use general rational functions instead of polynomial functions to approximate the response curve of the mean system time. This method seems very promising and the experimental results in Gong, Nananukul, and Yan [8] show that in many cases only a few light traffic derivatives can often yield very good approximations. The key condition needed in all the approximate methods mentioned above be that the mean system time is analytic at 0. So far, little is known about the analyticity of queueing systems. To this author's best knowledge, the only result on the analyticity is given by Zazanis [9], in which heproves performance measures defined on stochastic processes driven by Poisson inputs are analytic. His proof is based on a notion called absolute monotonicity. For the GI/G/ queue considered in this paper, his result in fact becomes a special case of ours. Usually, itisvery difficult to show analyticity. We needtoprove that () derivatives of arbitrary order exist and that (2) the corresponding MacLaurin/Taylor series converges to the correct value. The way we prove the analyticity can be briefly described as follows. We first construct a sequence of MacLaurin series based on a recursive procedure similar to the one derived in Gong and Hu [7]. (In fact, the one given there is a special case of ours.) We then prove that these MacLaurin series converge in a neighborhood of 0. Lastly, we show that

3 3 the moments of the system time are in fact equal to these MacLaurin series. So, we in fact prove that all the moments of the system time are analytic at 0. The key condition we use in establishing the analyticity is that the c.d.f. (cumulative distribution function) of the interarrival time can be expanded as a MacLaurin series over [0; ). This condition is satisfied by all phase-type distributions, but it excludes all c.d.f.'s whose support is not equal to [0; ), such asuniform and shifted exponential distribution functions. In fact, through two simple examples we illustrate that the mean system time is usually not analytic at 0 if the c.d.f. of the interarrival time does not have support [0; ). Therefore, the condition is perhaps also a necessary one though we are not able to prove it at this point. The rest of this paper is organized as follows. In Section 2, we present our main result. In Section 3, we discuss two examples, a uniform interarrival time and a shifted exponential interarrival time. We show that the analyticity at 0 does not hold for these two queues. The proof of our main result is given in Section. 2 The Main Result Consider a first-come-first-served GI/G/ queue with a renewal arrival process and i.i.d. (independently and identically distributed) service times. Let S be a generic service time and A be a generic interarrival time. Unless otherwise stated, throughout this paper we shall always assume E[S] < E[A] so that the queue is stable and ergodic. Denote the steady-state system time and waiting time of a job by T and W, respectively. (The system time is the time from when a job enters the system to when it leaves the system, and the waiting time is the time from when a job enters the system to when it enters service.) Let be the arrival rate. We assume A Y, where Y is a random variable independent of with E[Y ]. Then it is clear that T and W are functions of, as are their moments E[T n ]ande[w n ](n ; 2;:::). For ease of exposition, we willfocusont and W. Define T ~ T and W ~ W. Then T ~ and W ~ are the steady-state system time and waiting time of a job in the GI/G/ queue with interarrival time Y and service time S. Suppose Y has c.d.f. F ( ). First, we present our main result in the following theorem. Theorem. Suppose (A). All the moments of S are finite, and E[S n ]» ( ) n for n 0; ; 2;:::, where > 0

4 is a constant; (A2). F (x) is an analytic function over [0; ), and F (x) n0 F (n) (0 + ) x n for x 0, where F (n) (0 + ) is the n-th right-hand side derivative of F (x) at x 0 satisfying F (0 + ) < and jf (n) (0 + )j»(c F ) n (n ; 2;:::), where C F > 0 is a constant, then, E[ ~ T n ] and [ ~ W n ] (n ; 2;:::) are analytic at 0,or more precisely, there exists 0 > 0such that for 0»» 0 E[ ~ T n ] E[ ~ W n ] mn mn a nm m ; () b nm m : (2) Furthermore, coefficients d nm anm b nm and b nm can be calculated based on the following recursive equations, d nm b nm ( fin + P n P n ff 0 fi i n ib ii ; m n; fi i n ib i(m n+i) ; m>n; m n i0 ff i+ (d (n++i)m + b (n++i)m )+ ff 0 ff 0 d nm ; m n, () (3) for n ; 2;:::, where ff i F (i) (0 + )andfi i E[S i ]i! (i 0; ; 2;:::) The proof of Theorem will be given in Section. We should point out that a special case in which ff 0 0 is considered in Gong and Hu [7], where two recursive equations similar to (3) and () are derived for a nm and b nm (instead of d nm and b nm ); however, the analyticity is not discussed there. The way in which coefficients d nm and b nm are calculated via recursions (3) and () can be best illustrated by Table. We note in Table the numbers in the parentheses indicate the sequence in which thecorresponding calculations are performed.

5 6 Since E[T n ] E[ ~ T n ] n and E[W n ] E[ ~ W n ] n, it then immediately follows from Theorem that Corollary. Under the hypotheses of Theorem, E[T n ] and E[W n ] are analytic at 0. We note most service time distributions used in queueing theory, such as phase-type, uniform, and deterministic, satisfy Condition (A). Also the following proposition tells us that (A) is equivalent to the condition that S has finite moment generating function, i.e., there is a >0such that E[e S ] <. Proposition. (A) holds if and only if the moment generating function E[e S ] < for some >0. Proof. If (A) holds, choosing <,we then have K n0 for any K 0; ; 2;:::. Since E[S n ] n» n0 e S ( ) n n n0 S n n ; < ; by applying the dominated convergence theorem, we have E[e S ] <. On the other hand, if E[e S ] <, then, E[S n ] < E[e S ] n <. It is clear that the sequence f P K n0 E[S n ] n ;K 0; ; 2;:::g is monotonically increasing and bounded by E[e S ], hence its limit exists, and furthermore, by applying the dominated convergence theorem we have P n0 E[S n ] n <, hence (A) must hold. This completes the proof. 2 We now examine Condition (A2). Clearly, (A2) is satisfied by all phase-type distributions (see Neuts []). On the other hand, it excludes all c.d.f.'s whose support is not [0; ), such as uniform, triangle, and shifted exponential distributions. Two examples in the next section show that if Y has either a uniform distribution or a shifted exponential distribution, then E[T ] is not analytic at 0. Thus, it appears that (A2) might also be a necessary condition for E[T ] to be analytic at 0,although we are unable to provide a proof at this point. Before closing this section, we should point out that even though E[T n ]ande[w n ]may not be analytic at 0 if (A2) is not satisfied, they can still have derivatives of any order at 0under very mild conditions. In fact, it is shown in Hu [9] that if F (x) is analytic

6 7 at x 0 + (instead of [0; ) required in (A2)), then E[T n ] and E[W n ] have derivatives of any order at 0;alsoseethetwo examples given in the next section. Furthermore, these derivatives can be easily calculated through recursions (3) and (). 3 Two Negative Examples In this section, weprovide two examples in which E[T n ] and E[W n ] are not analytic function at 0; one is the U/M/ queue and the other is the (M+d)/M/ queue (where M+d represents the shifted exponential distribution, i.e., an exponential random variable plus a constant d). For illustrative purpose,we shall focus on E[T ]. We note for the GI/M/ queue (see Takacs [5]), E[T ] μ( ff) ; (5) where ff is the unique solution to the following equation in the range 0» ff< ff E[e μ( ff)y ]; (6) and μ is the service rate. In fact, T is exponentially distributed with rate μ( ff), thus E[T n ](E[T ]) n. Note that when 0,wehave E[T ]μ and ff 0,it then follows from (5) that E[T ] is analytic at 0if and only if ff is analytic at 0. Therefore, we only need to investigate analyticity of ff at The U/M/ Queue Suppose Y is uniformly distributed over [0; 2], i.e., f(x) 2 for 0» x» 2 and 0 otherwise, and S is exponentially distributed with μ. Then, (6) becomes Since ff 2( ff) e 2( ff) : (7) e 2( ff) lim 0 for n 0; ; 2;:::,!0+ n it is then not too difficult to verify based on (7) that the nth derivative of ff with respect to exists at 0(n ; 2;:::). Next we want to calculate d n ffd n j 0. Denote ff n d n ff d n fi fififi 0 for n ; 2;:::.

7 8 Rewrite (7) as 2ff( ff) e 2( ff) : (8) We note Differentiating (8) and letting 0, we have and ff 2. Proposition 2. Thus, Proof. Let us consider d n ( e 2( ff) ) d n fi fifififi 0 0; for n ; 2;:::. ff n ff n n k ff k ff n k ; for n 2; 3;:::, (9) (2n 2)! ; for n ; 2;:::. (0) 2 n (n )! d n ff d n fi fififi 0 n (2n 2)! 2 n (n )! : s 2 2 (2n 2)! 2 n (n )! n ; where 2 [0; 2). (Note: the power series converges when 0» <2.) We have s 2 + (2n 2)! 2 n (n )! n Hence, n n (2n 2)! 2 n (n )! k 2 s ψ n n n2 k (2k 2)! 2 k k!(k )! 2 A! 2 (2n 2)! 2 n (n )! n 2 (2k 2)! 2 k k!(k )! (2(n k) 2)! 2 n k (n k)!(n k )! (2(n k) 2)! 2 n k (n k)!(n k )! n : for n 2: () Clearly, (0) holds for n. By applying induction (0) then follows immediately from (9) and (). 2

8 9 Based on the proof of Proposition 2, we also have s ff n n 2 2 ; n which isobviously not equal to ff defined by (7). Therefore, ff is not analytic at 0. Since E[T ]( ff), E[T ] also has the n-th derivative at 0,and d n E[T ] d n fi fififi 0 (2n)! 2 n (n +)! ; (2) for n ; 2;:::. We note that (3) and () result in the same formula. In fact, we canalso use (2) to show that E[T ] is not analytic at The (M+d)/M/ Queue We now consider the (M+d)/M/ queue in which S is exponentially distributed with μ and f(x) ke k(x d) for x d and 0 otherwise, where 0» d< and k ( d). We then have from (6) ff k k + ff e ( ff)d : (3) Similar to the first example, we can show based on (3) that ff has derivatives of any order at 0, and Hence, d n ff d n fi fififi 0 0; n ; 2;:::: n which implies that ff is not analytical at 0. Proof of Theorem d n ff d n fi fififi 0 n 0; In this section, we prove Theorem. Since we consider exclusively the GI/G/ queue with interarrival time Y and service time S in this section, for simplicity of notation, we will use T and W (without tilde) to denote its system time and waiting time. Also, we sometimes may write T and W as T ( ) andw ( ) to emphasize their dependence on. Without loss of generality, we assume C F is small enough such thatc F < and C F C F + ff 0 < : ()

9 0 (Recall ff 0 <.) Otherwise, we consider a GI/G/ queue with interarrival time CY and service time C S, where C>C F is a constant such that C F C C F + ff 0 < : The system time and waiting time of this GI/G/ queue are CT and CW, respectively. The c.d.f. of CY is ~ F (x) F (xc). It is clear that ~ F (x) is also analytic over [0; ); furthermore, ~ F (0) F (0) < and j ~ F (n) (0)j jf (n) (0)C n j»(c F C) n. If E[(CT) n ] and E[(CW) n ] are analytic at 0, then so are E[T n ]ande[w n ]. Lemma. T ( ) has finite moment generating function. Then according to Proposition, there exists a constant C T ( ) > 0 (depending on ) such that E[T ( ) n ]» (C T ( )) n. Furthermore, there is > 0 such that C T ( ) < for0»». Proof. The first part of lemma falls out directly from Thorisson [6]; also a different proof can be found in Wolff [8]. We now consider the second part of the lemma. From Weber [7], we know that T ( ) is a.s. a convex function of. Also note T (0) 0, therefore, T ( )» T () a.s. for 0»». We choose such that C T () <. Then, E[T ( ) n ]» ( C T ()) n for 0»». 2 In the rest of this section, we will restrict ourselves to 2 [0; ]. It is well known that T and W satisfy T W + S (5) and W d max(t Y;0); (6) where d means equal in distribution. We note W and S in (5) are independent of each other, as are T and Y in (6). Then it immediately follows from (5) that E[T n ] n E[S n k ] (n k)! E[W k ] n k : (7) k! On the other hand, (6) gives E[W n ] E E "Z T 0 "Z T 0 (T x) n df (x) (T x) n # F (k+) (0 + ) x k dx + F (0 + ) T n k! # :

10 Since Z T 0 fi fi fi fi fi (T x)n fi F (k+) (0 + ) fifififi x k dx» k! < Z T 0 (T x) n x k (C F ) k+ dx k! T n+k+ (n + k +)! (C F ) k+ and (note C F E[T n+k+ ] (n + k +)!» (C T ) n+k+ < < and C T ( ) < for 0»» ), by applying the dominated convergence theorem twice (one for interchanging the summation and the integration and the other for interchanging the summation and the expectation), we have E[W n ] F (k+) (0 + ) E[T n+k+ ] (n + k +)! + F (0+ ) E[T n ] F (k) (0 + ) E[T n+k ] (n + k)! : (8) Substituting (7) into (8), we obtain the following infinite system of linear equations for fe[w n ];n; 2;:::g E[W n ] c n + n+k ff k i0 E[W i ] fi n+k i n+k i i! ff k fi n+k n+k + j n ; 2;:::. (Recall ff i F (i) (0) and fi i E[S i ]i!.) n+k ff k i E[W i ] fi n+k i n+k i i! E[W j ] c nj ; (9) j! We first want toshow that the infinite system of linear equations defined by (9) has a unique solution. We use the following result from Kantorovich and Krylov [0]. Lemma 2. Consider the infinite system of linear equations 2 6 c c 2 c 3 ::: c 2 c 22 c 23 ::: c 3 c 32 c 33 ::: z z 2 z c c 2 c ;

11 2 where c n 's and c nj 's are real numbers satisfying jc n j»m and j jc nj j»m 2 < ; (n ; 2;:::) where M and M 2 are two constants. This system has one and only one bounded solution fzn; Λ n ; 2;:::g. Furthermore, if fzn N ; n ; 2;:::;Ng is the solution of the finite system of linear equations 2 6 c c 2 ::: c N c 2 c 22 ::: c 2N z z c c c N c N 2 ::: c NN z N c N then fzn N ; n ; 2;:::;Ng is uniformly bounded, i.e., jzn N j <M for some constant M, and z Λ n lim N! zn n ; (n ; 2;:::;): Proof. See Kantorovich and Krylov [0] pp Lemma 3. There is 2 > 0 such that for every 2 [0; 2 ] the infinite system of linear equations defined by (9) has one and only one bounded solution. Proof. To apply Lemma 2, we only need to verify jc n j»m and j Based on (), we can choose 2 2 (0; ]suchthat 2 < and Then, if 2 [0; 2 ], we first have jc n j» Secondly, wehave jff k jfi n+k n+k» j jc nj j» k jc nj j»m 2 < (n ; 2;:::): 2 CF C F + ff 0 < : (C F ) k ( 2 ) n+k + ff 0 ( 2 ) n» C F + ff 0 < : C F n+k jff k j i fi n+k i n+k i n+k n» (C F ) k s 2 ) k i(c n+k i + ff 0 ( 2 ) n i i CF» + ff 2 C 0 F < :

12 3 This completes the proof. 2 Lemma. Suppose d nm 's and b nm 's are defined by recursive equations (3) and (). We have jd nm j»(c ) m and jb nm j»(c ) m C ; (20) for m ; 3;:::, and n ; 2;:::;m,whereC is any constant satisfying C > 2, and Proof. Since C > 2, and it is not difficult to show that and C C CF C F + ff 0 < : C C CF C F + ff 0 < ; < C ; (2)» C F C + ff 0 ff 0 C F < C : (22) We now use induction to prove (20). It is obvious that (20) holds for m. Suppose (20) holds for m» k. We now consider the case m k +. For n k +,wehave and jd k+;k+ j» jfi k+ j +» ( ) k+ + k i fi k+ i jb ii j k i» (C ) k+ C ( ) k+ i (C ) i C k i0» (C ) C k+ C (C ) k+ ; Cs C k+ i jb k+;k+ j ff 0 ff 0 jd k+;k+ j»(c ) k+ C : Hence, (20) holds for n k +. We assume that (20) holds for n l +;:::;k +. Therefore, for n l we have jd l(k+) j» l i jfi l i jjb i(k+ l+i) j

13 and» l i ( ) l i (C ) k+ l+i C (C ) k+ C l i» (C ) C k+ C (C ) k+ ; ( C ) l i " k l jb l(k+) j» jff i+ j(jd ff (l++i)(k+) j + jb (l++i)(k+) j)+ff 0 jd l(k+) j 0 i0 "» (C F ) k+ k l (C F ) + C # i+ + ff 0 ff 0 C i0 s»» (C ) k+ CF C + ff 0 ff 0 C F» (C ) k+ C : (The last inequality is based on (22).) This completes the proof. 2 We are now ready to prove Theorem. Proof of Theorem. From Lemma, we have # ja nm j»(c ) m C and jb nm j»(c ) m (C ) : Hence there exists 0 2 (0; 2 ] such that for any 2 [0; 0 ] the following two MacLaurin series fi n ( )! n ( ) mn mn a nm m b nm m (n ; 2;:::) converge, and fi n ( )» C and! n ( )» (C ). Since a nm 's and b nm 's are defined (through d nm 's and b nm 's) by recursive equations (3) and (), it is not difficult to verify that for 0»» 0 f! n ( );n; 2;:::g satisfies! n ( ) ff k fi n+k n+k + n+k ff k fi n+k i! i ( ) n+k i : i

14 5 We can also verify fi n ( ) n fi n k! k ( ) n k : Therefore, both fe[w ( ) n ];n ; 2;:::g and f! n ( );n ; 2;:::g are bounded solutions to the infinite system of linear equations defined by (9), which according to Lemma 3 has only one bounded solution. This immediately leads to E[W ( ) n ] And we alsohave E[T ( ) n ] mn mn b nm m ; for 0»» 0. a nm m ; for 0»» 0. This completes the proof. 2 Acknowledgement This research is partially supported by the National Science Foundation under Grants EID and DDM References [] Asmussen, S., Light Traffic Equivalence in Single Server Queues," to appear in The Annals of Applied Probability. [2] Benes, V., Mathematical Theory of Connecting Networks and Telephone Traffic, Academic Press, New York, 965. [3] Burman, D.Y. and Smith, D.R., A Light-Traffic Theorem for Multi-Server Queues," Math. Oper. Res., 8, pp.5-25, 983. [] Burman, D.Y. and Smith, D.R., An Asymptotic Analysis of a Queueing Systems with Markov-Modulated Arrivals," Operations Research, 3, pp.05-9, 986. [5] Daley, D.J. and Rolski, T., Light Traffic Approximation in Queues," Mathematics of Operations Research, Vol. 6, pp. 57-7, 99. [6] Fendick, W. and Whitt, W., Measurements and Approximations to Describe the Offered Traffic and Predict the Average Workload in a Single-Server Queue," Proceedings of IEEE, Vol.77, No., pp.7-9, 989.

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