The Distribution of the Number of Arrivals in a Subinterval of a Busy Period of a Single Server Queue

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1 The Distribution of the Number of Arrivals in a Subinterval of a Busy Period of a Single Server Queue A. Novak (a.novak@ms.unimelb.edu.au) Department of Mathematics and Statistics, University of Melbourne, Australia P. Taylor (p.taylor@ms.unimelb.edu.au) Department of Mathematics and Statistics, University of Melbourne, Australia D. Veitch (d.veitch@ee.unimelb.edu.au) ARC Special Research Center for Ultra-Broadband Information Networks (CUBIN), an affiliated program of National ICT Australia, Department of Electrical and Electronic Engineering, University of Melbourne, Australia September 16, 25 Abstract. In the course of attempting to estimate the arrival rate of a single server queue using an active probing experiment, the authors found it necessary to derive the distribution of the number of arrivals between two probes under the conditions that the busy period of the queue lasts this long. In this paper we derive this distribution. The key building blocks in the derivation of the distribution are the classical ballot theorem and its generalized forms. Keywords: active probing; ballot theorem; busy period; M/D/1; M/G/1 1. Introduction Much is known about the single server queue. In particular, we can find in the literature a variety of approaches to deriving the distribution of the number served in the busy period. Lajos Takács was one of the first to do so, using a generalization of the classical ballot theorem [6]. Later on, Kleinrock used a different approach in which the order of customers served is permuted to achieve an FCLS queueing discipline [5]. In this paper we derive a related distribution: that of the number served in a subinterval of a busy period, drawing on Takács ideas of the ballot theorem and its generalization. The motivation for this problem surfaced as a result of developing a packet-pair based method used to estimate the arrival rate of Internet traffic, on a single link. Such a packet-pair based method is one of the standard active probing techniques. Active probing involves releasing streams of packets across the network to measure network properties. It has become one of the main c 25 Kluwer Academic Publishers. Printed in the Netherlands. SubintBP.tex; 16/9/25; 14:7; p.1

2 2 A. Novak, P. Taylor and D. Veitch Figure 1. Basic active probing infrastructure sources of information about Internet behaviour, and the object of several large scale measurement activities (for example see [12]). One of the pioneering works on active probing, in particular packet-pair probing, is the thesis of Keshav [9]. Active probing can be used to measure a number of different quantities, such as packet loss across a network path, the capacities of links [2, 1], and the unused capacity or available bandwidth across a path [11]. However, there is much room for improvement in existing techniques, which are typically based on heuristics, and lack a well defined probabilistic description. Such a foundation is necessary to examine the operation of these methods in more detail, to determine their statistical performance, and to explore possible optimisation. Figure 2. Timestamps The incentive for the problem discussed in this paper stems from the following packet-pair experiment. It basically comprises a Sender injecting a stream of scheduled and controlled pairs of packets, called Probes, into the network, which traverse some route in the network and terminate at the Receiver (see Figure 1). We set the separation time of the probes within each probe-pair as they arrive to the network, and measure the output separation times of the same pairs as they exit the SubintBP.tex; 16/9/25; 14:7; p.2

3 Distribution of the Number of Arrivals 3 network (refer to Figure 2). The objective is to estimate the arrival rate of cross traffic (CT), that is all non-probe traffic. A probabilistic analysis of a full network, with all the complexity of traffic and packet size processes fully described, is likely to be intractable. Thus, to make a start on gaining insight into the experiment described above, we consider a substantially simplified system. First, we reduce the network to a single hop. Second, we replace a realistic arrival process with a Poisson process. Third, we assume that packets have identical size x c. Figure 3. Single hop: FCFS queue. We inject a sequence of packet-pair probes into the network, such that the time separation between probes is d. Moreover, we choose parameter d to be an integer multiple of x c. The outcome of the experiment is a sequence of output separation times, τi. A little thought tells us that, if the probes comprising a probepair, share the same busy period, then the corresponding normalized output separation time measurement, τi /x c, will be exactly equal to the number of packets that have arrived in the time interval, d. If both probes arrive to see an empty system, the output separation will be the same as the input separation d. Otherwise the corresponding output separation measurement will (with probability 1) not be equal to an integer multiple of x c. Figure 4 depicts a histogram of the output separation times produced by a simulation experiment in which d =.9 and x c =.1. The peaks in the histogram occur when the output separation times are exactly equal to an integer multiple of the CT-packet service time. That is, as a result of probes sharing the same busy period or both arriving to an empty system. On the other hand, the remaining output separation times, τi, are not an integer multiple of the CT-packet service time, SubintBP.tex; 16/9/25; 14:7; p.3

4 4 A. Novak, P. Taylor and D. Veitch 4 35 Number of PB Packet Pairs [#] NOISE PEAKS Output Separation Time [sec] Figure 4. Histogram of output separation times. System utilization of 75% and input separation of.1 sec. x c. We denote the collection of such output separation times as noise in the histogram. One way to obtain the information about peaks from the above experiment would be to filter out all the data corresponding to noise. Filtering is performed by simply discarding all measurements which are not equal to an integer multiple of x c. We are seeking the exact distribution of the remaining output separation times, which translates to the number of cross traffic packets arriving between the probes, given that the busy period lasts that long. Here, we shall consider an abstract version of this problem, which is independent of probes. Let t 1 and t 2 be two arbitrary time points within a busy period such that t 2 t 1 is an integer multiple of x c. Then, we are interested in calculating the distribution of the number of packets that arrive from time t 1 until time t 2, conditional on the busy period lasting that long. In this abstract version of the problem, our techniques apply to the M/G/1 generalization of the M/D/1 queue. We prove our results first in the M/D/1 context and the M/G/1 result will be stated at the end. As we mentioned above, in this paper we will make use of the classical ballot theorem. In fact, the ballot theorem has been used to solve various other queueing theory problems. Two recent examples are He and Sohraby [4], who obtained the closed form analytical expressions for the stationary distribution of the system queue lengths, and Gure- SubintBP.tex; 16/9/25; 14:7; p.4

5 Distribution of the Number of Arrivals 5 witz, Sidi and Cidon [8], who derived an explicit expressions for the distribution of the number of lost packets within a packet block of consecutive packet arrivals into a finite buffer. 2. The Distribution of the Number of Arrivals in a Subinterval of a Busy Period of an M/D/1 queue Consider an M/D/1 queue with the following characteristics: One server is available and the service is offered in a FCFS order. The server processes the work load at unit rate. Packets arrive in a Poisson stream with parameter λ. Hence, the utilization of the system is ρ = λx c. Note that we have not, at this stage, made the assumption that ρ < 1. Let s be a positive integer. Then, we are interested in calculating the distribution of the number of packets that arrive from time t until time t + sx c, conditional on the busy period lasting that long. The system is assumed busy at time t, with l 1 packets in the system. Let us denote by ν r the number of packet arrivals during the service of the r th packet, relative to t, in a busy period. Thus, the cumulative number of arrivals up to the conclusion of the r th packet s service is N r, where N r ν 1 + ν ν r and N =. If we denote by B the event that the busy period lasts longer then sx c time units, than we are interested in finding P[N s = k B] The Time Period Begins with a Service Completion Let t be a time point at which a service finishes. In this subsection we derive the probability that k packets arrive in an interval of the form (t,t+sx c ) and that the busy period lasts for this time, given that l 1 packets are present in the system (the server plus the queue) at time t. This distribution is given by the following theorem. SubintBP.tex; 16/9/25; 14:7; p.5

6 6 A. Novak, P. Taylor and D. Veitch Theorem 2.1. Let B denote an event that the busy period lasts more then s service times. Then, for ρ R +, s N + and l N +, (sρ) k e sρ, k, l > s k! fs,ρ(k,b l) d = (1) where [ h d s,ρ(k,b l) = ρk k! e sρ s k h d s,ρ (k,b l), k > (s l)+, 1 l s, k (s l) + s l i= ( ] k l )(i + l) i 1 (s l i) k i. (2) i Proof. The proof is simple when l > s, since in this case the busy period is sure to last at least sx c -time units. Hence, f d s,ρ (k,b l) = P[N s = k], k. (3) However, when 1 l s we do not have the same simplification. Here we also have to ensure that the busy period will last over the entire interval of interest, that is f d s,ρ (k,b l) = P[N r > r l for r = 1,2,...,s and N s = k] (4) for k > {s l} + and otherwise. Using a version of the ballot theorem discussed in [6], equation (4) can be rewritten as follows f d s,ρ(k,b l) = P[N s = k] j=l l j P[N j = j l]p[n s j = k (j l)]. (5) To prove this, it is sufficient to show that the sum in the right hand side of equation (5) is the probability that N r r l for some r = 1,2,...,s and N s = k. This event occurs in the following mutually exclusive ways: The smallest r for which N r = r l is r = j {l,...,s}. Then, N j = j l and N r > r l for r = 1,2,...,j 1 or equivalently, N j N r < j r for r = 1,2,...,j 1. Let us condition on the event that N j = j l and N s = k. That is, we shall consider P[N j N r < j r for r = 1,2,...,j 1 N j = j l,n s = k]. (6) r Now, N r = ν i for r = 1,...,s by definition and the ν i are independently and identically distributed. Therefore, (6) is the i=1 same SubintBP.tex; 16/9/25; 14:7; p.6

7 Distribution of the Number of Arrivals 7 Figure 5. The unfinished work, η(t) vs. normalized time, where x c is the packet service time. Circles denote integer multiples of the service time. Thus, n i = η(ix c) and specifically n 1 = l+ν 1 1 = l+n 1 1, n 2 = l+(ν 1 1)+(ν 2 1) = l+n 2 2,..., n s = l + N s s = l + k s. In order for the busy period to last longer than sx c-time units, we need n i > for i = 1,2,..., s. as [ j P i=r+1 ν i < j r for r = 1,2,...,j 1 ] j ν i = j l, ν i = k. i=1 i=1 (7) Introducing a time reversing change of index ν i ν s i+1 so that ν 1 ν s and ν s ν 1, (7) becomes [ ] s r P ν i < j r for r = 1,2,...,j 1 ν i = j l, ν i = k i=s j+1 i=s j+1 (8) and since ν i are i.i.d. and consequently cyclicly interchangeable, this is equal to [ j r ] j P ν i < j r for r = 1,2,...,j 1 ν i = j l, ν i = k. (9) i=1 The time reversed vector { ν 1... ν j 1 } is distributed the same way as {ν 1... ν j 1 } and so returning to the more compact notation, (9) can i=1 i=1 i=1 SubintBP.tex; 16/9/25; 14:7; p.7

8 8 A. Novak, P. Taylor and D. Veitch be expressed as P[N j r < j r for r = 1,2,...,j 1 N j = j l,n s = k] = P[N r < r for r = 1,2,...,j 1 N j = j l,n s = k]. This is equal to 1 j l = l j j by the ballot theorem [6, pp.1]. Furthermore, (1) (11) P[N j = j l,n s = k] = P[N j = j l,n s N j = k (j l)] = P[N j = j l]p[n s N j = k (j l)] = P[N j = j l]p[n s j = k (j l)] by the independence of ν i. If we multiply equation (11) by P[N j = j l]p[n s j = k (j l)] and sum over all l j s, then we get the sum in equation (5). Hence, the distribution we are seeking is P[N s = k], k, l > s where fs,ρ d (k,b l) = h d s,ρ(k,b l) = P[N s = k] h d s,ρ (k,b l), k > (s l)+, 1 l s, k (s l) + j=l (12) l j P[N j = j l]p[n s j = k (j l)]. (13) Since the arrival process is Poisson and service time is x c for all packets, then in general P [N r = j] = (λrx c) j j! e λrxc = (rρ)j e rρ. (14) j! Substituting this into the right-hand side of equation (13) we get [ h d s,ρ(k,b l) = ρk k! e sρ s k = ρk k! e sρ ( ] k l )(j) j l 1 (s j) k (j l) j l j=l [ ( ] s l s k k l )(i + l) i 1 (s l i) k i i i= (15) SubintBP.tex; 16/9/25; 14:7; p.8

9 Distribution of the Number of Arrivals 9 for k > (s l)+. Finally, we complete the proof by substituting equation (15) into equations (3) and (5) The Time Period Begins at an Arbitrary Point Within a Busy Period In this subsection we extend the result of Theorem 2.1 by considering a subinterval (t,t + sx c ), where t is an arbitrary time point within a busy period, rather then a point of service completion. Let f c s,ρ(n,b l) be the probability that n packets arrive in the time interval (t,t+sx c ) and the busy period lasts longer then s service times, conditional on l packets being present in the system at time t. Theorem 2.2. For ρ R +, s N + and l N +, f c s,ρ(n,b l) = ϕ = n ϕ k={s l ϕ +1} + f d s 1,ρ(k,B l + ϕ 1)θ ρ (n k) + ϕ =1 n ϕ k={s ϕ } + f d s 1,ρ (k,b ϕ )θ ρ (n k) (16) for n (s l) + and otherwise. The distribution function for fs 1,ρ d (k,b ) is given in Theorem 2.1 and θ ρ (σ) = 1 σ + 1 ρσ σ! e ρ. (17) Proof. Let V t be the time measured from t until the server becomes free to serve another packet. Thus V t is the remaining service time of the initial service. Similarly, sx c -time units later there is a certain number of packets present in the queue. Let U t+sxc be the time measured backwards from t + sx c to the last service completion at or before t + sx c. Thus, U t+sxc is the age of the packet in use at time t + sx c. From Figure 6, it is obvious that U t+sxc = x c V t. Hence, by knowing the distribution of the residual life of the packet in service at time t, that is V t, we can directly find the distribution of the age of the packet in use at time t+sx c, that is U t+sxc. Let g(ϕ,ϕ e ) denote the probability that there are ϕ and ϕ e arrivals in times V t and U t+sxc, respectively. For n packets to arrive in the time that it takes to service s packets, ϕ must arrive in the time interval V t, ϕ e during time U t+sxc and the SubintBP.tex; 16/9/25; 14:7; p.9

10 1 A. Novak, P. Taylor and D. Veitch Figure 6. Diagram of cumulative number of packet arrivals, N t versus time. On the time axis, V t represents the forward recurrence time to the end of the service in progress at time t and U t+sxc represents the backward recurrence time to the beginning of the service in progress at time t + sx c. On the vertical axis, ϕ is the number of packets that arrive during the time V t and ϕ e is the number of packets that arrive during the time U t+sxc. rest during the remaining (s 1)x c units of time (see Figure 6). If there is more than one packet in the system (that is l > 1) then, it is certain that the busy period will continue after the end of the current packet s service. For n (s l) +, l > 1 we can write f c s,ρ (n,b l) = n ϕ = n ϕ k={s l ϕ +1} + f d s 1,ρ (k,b l +ϕ 1)g(ϕ,ϕ e ), (18) where ϕ e = n k ϕ. However if l = 1, the busy period will continue only if at least one packet has arrived during V t. Thus, for n (s l) + f c s,ρ(n,b 1) = ϕ =1 n ϕ k={s ϕ } + f d s 1,ρ(k,B ϕ )g(ϕ,ϕ e ). (19) Because the forward recurrence time has a uniform distribution on the interval [,x c ], the probability of joint number of arrivals in times V t SubintBP.tex; 16/9/25; 14:7; p.1

11 Distribution of the Number of Arrivals 11 Figure 7. The unfinished work, η(t) vs. normalized time. Circles denote integer multiples of the service time and δ = V t/x c. Thus, n i = η(ix c + V t) and specifically n = l + ϕ 1 + N = l + ϕ 1,..., n r = (l + ϕ 1) + ν 1 + ν ν r r = (l + ϕ 1) + N r r etc. and η(s) = (l +ϕ 1)+N s 1 (s 1)+ϕ e = l + n s. In order for the busy period to last at least until the arrival of the second probe, we need n i > for i =, 1,..., s 1 and n (s l) +. and U t+sxc can be found by evaluating the integral g(ϕ,ϕ e ) = xc (λv t ) ϕ ϕ! = λϕ +ϕ e ϕ!ϕ e! e λxc x c e λvt (λ(x c v t )) ϕe e λ(xc vt) ϕ e! xc v ϕ t (x c v t ) ϕe dv t. 1 x c dv t If we substitute v t = ux c, we can easily recognize the integral as a beta function. Hence, xc 1 v ϕ t (x c v t ) ϕe dv t = x ϕ +ϕ e+1 c u ϕ (1 u) ϕe du = x ϕ +ϕ e+1 ϕ!ϕ e! c (ϕ + ϕ e + 1)! SubintBP.tex; 16/9/25; 14:7; p.11

12 12 A. Novak, P. Taylor and D. Veitch and therefore g(ϕ,ϕ e ) = 1 ϕ + ϕ e + 1 ρ ϕ +ϕ e (ϕ + ϕ e )! e ρ. (2) Since g(ϕ,ϕ e ) is a function of ϕ +ϕ e, we can write g(ϕ,ϕ e ) = θ ρ (ϕ + ϕ e ). If the sum ϕ + ϕ e = σ, then θ ρ (σ) = 1 σ + 1 ρσ σ! e ρ. (21) Theorems 2.1 and 2.2 both hold for any ρ R +. However, in the special case when ρ < 1, we can say something about the stationary distribution of the number of packets that arrive in a subinterval of a busy period. Corollary 2.1. When ρ < 1 the stationary probability that n packets arrive in time interval (t,t+sx c ) and the busy period lasts longer than s service times, is f c s,ρ(n,b) = where, and l=1 ϕ =I{l=1} n ϕ k={s l ϕ +1} + f d s 1,ρ(k,B l+ϕ 1)θ ρ (n k)p l P = Q P l = Q l Q l 1, l 1 + (1 Q s ) (sρ)n e sρ (22) n! (23) l Q l = 1 ρ ( 1) j e ρ(l j)[ρ(l j)]j, l. (24) j! j= Proof. If we remove the conditioning on l in equation (16), we get fs,ρ c (n,b) = fs,ρ c (n,b l)p l = l=1 l=1 ϕ =I{l=1} n ϕ k={s l ϕ +1} + f d s 1,ρ (k,b l + ϕ 1)θ ρ (n k)p l (25) for n (s l) + and otherwise; where I {.} is the indicator function and P l is the probability that l packets are present in the system at time t. SubintBP.tex; 16/9/25; 14:7; p.12

13 Distribution of the Number of Arrivals 13 We can further simplify the above probability mass function, f c s,ρ(n,b) such that it is easily computable. When l > s, it follows that l+ϕ 1 s and hence we can write f c s,ρ(n,b) = l=1 ϕ =I{l=1} + n ϕ fs 1,ρ(k,B l+ϕ d 1)θ ρ (n k)p l k={s l ϕ +1} + n ϕ fs 1,ρ(k,B)θ d ρ (n k)p l (26) l=s+1 ϕ = k= where in the right summand fs 1,ρ d (k,b) is now independent of l, by equation (1). Therefore, f c s,ρ(n,b) = l=1 ϕ =I{l=1} + n ϕ fs 1,ρ(k,B l+ϕ d 1)θ ρ (n k)p l k={s l ϕ +1} + n ϕ fs 1,ρ d (k,b)θ ρ(n k) P l. (27) ϕ = k= l=s+1 By [6, pp.29 (12)], P l is given by (23), where Q l is given by (24) l=s+1 P l = 1 P l = 1 Q s. (28) l=1 Substituting this into equation (27) and expanding f s 1,ρ d(k,b) and θ ρ (n k) in the right-hand summand of equation (27), we get f c s,ρ(n,b) = l=1 ϕ =I{l=1} + (1 Q s ) ϕ = n ϕ k={s l ϕ +1} + f d s 1,ρ(k,B l+ϕ 1)θ ρ (n k)p l n ϕ k= ((s 1)ρ) k e (s 1)ρ k! ρ n k (n k)! e ρ, (29) which can then be easily simplified to yield the result stated in equation (22) and thus complete the proof. This result is in line with our intuition, since when l > s we can guarantee that the busy period will last during the entire length of the subinterval and the number of packet arrivals will follow a Poisson distribution with parameter sρ. Let f c s,ρ(n B) be a probability that n packets arrive in the time interval (t,t+sx c ) given that the busy period lasts longer then s service times. SubintBP.tex; 16/9/25; 14:7; p.13

14 14 A. Novak, P. Taylor and D. Veitch Corollary 2.2. For ρ < 1, s N + and l N +, where fs,ρ c (n,b) is given in Corollary 2.1 and P[B] = n= l=1 ϕ =I{l=1} fs,ρ(n B) c = fc s,ρ(n,b), (3) P[B] n ϕ k={s l ϕ +1} + f d s 1,ρ(k,B l+ϕ 1)θ ρ (n k)p l + (1 Q s ). (31) Proof. The probability that the busy period lasts longer then s service times is obtained simply by summing equation (22) for all possible values of n. 3. Properties of the Distribution of the Number of Arrivals in a Subinterval of a Busy Period The parameters that determine the shape of the distribution of the number served in a subinterval of the busy period are s and ρ s = 3 s = 1 s = n n n Figure 8. The probability mass function of the number of arrivals in a subinterval of a busy period, f c s,ρ(n B). The utilization for all three curves is ρ =.5 and the probe separation (left to right) is s = 3, s = 1 and s = 17. The former is responsible for the right shift along the x-axis. That is, as s increases, so does the shift and vice versa. This is in line with our SubintBP.tex; 16/9/25; 14:7; p.14

15 Distribution of the Number of Arrivals 15 intuition. The parameter s denotes the input-separation of the probes in terms of whole packets. If n < s, then the busy period will last for s service times only if the number l, of packets present when the first probe arrives is large, which is rare. That is, when n < s, the probability that n packets arrive in a subinterval of the busy period is low. We observe this behavior on the graph, as a right shift along the x-axis (see Figure 8) f c 3,ρ (n B) n 1 15 ρ Figure 9. The probability mass function of the number of arrivals in a subinterval of a busy period ploted as a function of ρ and n. The probe separation is s = 3 service times. On the other hand, ρ determines the shape of the curve. In the physical world ρ represents the system utilization and for a stable system takes on values < ρ < 1. When ρ is smaller, the distribution is narrower and taller and, when ρ is larger, the opposite is true. This is illustrated in Figure 9. The intuition behind this observation is that if ρ is small, the probability (31) that the busy period will last for s service times is smaller; but if it does, it will be concentrated around s. The reason for this is that it is a rare event for either the number of packets initially in the queue or the number arriving between the probes to be large. If we compare the distribution of the number served in a subinterval of a busy period with data obtained from a simulation, we see that we have a good match as expected (see Figure 1). SubintBP.tex; 16/9/25; 14:7; p.15

16 16 A. Novak, P. Taylor and D. Veitch Number of CT packets caught by the packet pair [pkt] Figure 1. Comparison of simulated data and actual pdf f c s,ρ(n B). The vertical lines are obtained by creating a histogram from the simulated data. The height of the line is its probability. The dots connected via a dashed line represent the evaluations of the probability density function f c s,ρ(n B). Relevant parameters: ρ =.85, s = 3 and the total number of probe-pairs was 2,. 4. The Distribution of the Number of Arrivals in a Subinterval of a Busy Period of an M/G/1 queue If only momentarily, we eliminate a time dimension and consider the problem in terms of packets, we see that the analysis for M/D/1 and M/G/1 queues are identical. Thus, we notice that results of the previous section apply just as well to M/G/1 queue. We should note however, that the result gives us the distribution of the number of packets that arrive during the service of the s packets, rather then in a fixed time. Thus the usefulness of this result is less clear. Consider an M/G/1 queue with the following characteristics: There is one server available and the service is offered in a FCFS order. The service times, χ 1,χ 2,... are mutually independent and identically distributed positive random variables with a distribution function B(x). Packets arrive in a Poisson stream. SubintBP.tex; 16/9/25; 14:7; p.16

17 Distribution of the Number of Arrivals 17 For a positive integer s, we are interested in calculating a distribution of the number of packets that arrive in M/G/1 queue, in the time interval it takes to service s packets, conditional on the busy period lasting that long. The majority of analysis we presented for M/D/1 queue still holds in an M/G/1 queue. For example, the probability mass function for the case when the start of the subinterval overlaps with a service completion, given by equation (12), also holds for an M/G/1 queue. However since the service times are no longer equal for all packets, the probability of the cumulative number of arrivals will now differ from the M/D/1 queue. The correct probability for the cumulative number of arrivals up to the r th packets service, in an M/G/1 queue is given by P[N r = j] = e λx(λx)j db r (x), (32) j! where B r (x) is the distribution of the time taken by r successive services. B r (x) is clearly the r-fold convolution of the service time distribution B(x). This leads to the following theorem. Let t be a time point at which a service finishes. We wish to derive the probability that k packets arrive during the time it takes to complete s services and that the busy period lasts for this time, given that l 1 packets are present in the system at time t. Theorem 4.1. For ρ R +, λ >, s N + and l N +, f g,d s,ρ (k,b l) = e λx xk k! db s(x), k, l > s h g s,ρ(k,b l), k > (s l) +, 1 l s, k (s l) + (33) where h g s,ρ(k,b l) is ( s l 1 e λx (λx) i i! i= [ λ k a k s k! j=l ) and a j r = e λx x j db r (x). ( ) l k j j l db s (x) j=l a j l j a k j+l s j [ l j aj l j ] i=s l+1 ( ) ] i a i j+l s j j l (34) We can now extend the result from Theorem 4.1 to hold when t is an arbitrary time point within a busy period of the M/G/1 queue. To SubintBP.tex; 16/9/25; 14:7; p.17

18 18 A. Novak, P. Taylor and D. Veitch derive this distribution, we can apply the same principles as we did in equation (25). If we do so, and also remove the conditioning on the number of packets present in the system at time t, we obtain a result outlined in the theorem below. Theorem 4.2. For ρ < 1, λ >, s N + and l N +, f g,c s,ρ(n,b) = l=1 ϕ =I{l=1} n ϕ k={s l ϕ +1} + f g,d s 1,ρ (k,b l+ϕ 1)ĝ(ϕ,ϕ e )P l (35) for n (s l) + and otherwise; where ϕ e = n k ϕ. Here, the joint probability of the number of arrivals during time V t1 and U t2 is ĝ(ϕ,ϕ e ) = (λv t1 ) ϕ ϕ! e λvt 1 (λu t2 ) ϕe ϕ e! 1 B(x) e λut 2 dv t1 u t2 E[χ] where V t1 is the remaining service time of the initial service and U t2 is the age of the packet in use at time t 2. The probability P l is given by P l = Q l Q l 1, l 1 P = Q (36) where Q l = 1 Q P[N j = j + k], l 1 j=1 Q = 1 ρ. (37) 5. Estimating the CT packet rate In section 2 we derived the distribution of the number of packets served in a subinterval of a busy period of an M/D/1 queue. This problem was abstracted from the authors research in the area of active probing. In this active probing problem we are interested in learning about the specific network conditions, such as CT arrival rate, by using probes as the network explorers and recording their experience. In the experiment as conducted by the authors, only the experience of the probes that happen to share the same busy period is meaningful for further SubintBP.tex; 16/9/25; 14:7; p.18

19 Distribution of the Number of Arrivals 19 analysis. Therefore, the distribution we derived in section 4 models the meaningful probe experience. In the active probing context we are interested in estimating the parameter ρ from the observed data. We shall defer a detailed description of how best to do this to a later paper. However, to demonstrate the feasibility of such an estimation, we present some preliminary results here. Although complicated, the likelihood function calculated from the output data of an active probing experiment is a function of the single parameter ρ. It is relatively easy to use, for example, a Golden Section search to maximize the log-likelihood. Below is an example of one such experiment. Example 5.1. In this series of experiments a packet-pair stream was created such that the probe separation was s = 2x c and the probe size was. We ran a Matlab simulation in which we injected varying numbers of probe pairs into an M/D/1 queue, experiencing cross traffic with ρ =.75. Table I. Finding ˆρ, via the maximum likelihood method. n ˆρ 1 (.545,.968 ) 5 (.6371,.843 ) 1 (.6853,.8328 ) 5 (.7249,.791 ) 1 (.733,.7768 ) Using the maximum likelihood method we found an estimate ˆρ for ρ from each of these samples. We used the Golden Section Search to find the maximum value of the likelihood function, with the error set to 1 4. The confidence limits were obtained using the likelihood ratio. We see that it is possible to use the distribution derived in this paper to estimate ρ. SubintBP.tex; 16/9/25; 14:7; p.19

20 2 A. Novak, P. Taylor and D. Veitch Acknowledgements The first author would like to acknowledge the support of a Melbourne Research Scholarship. The third author was supported by the Australian Research Council. Also, all three authors acknowledge funding from the Australian Research Centre of Excellence in the Mathematics and Statistics of Complex Systems (MASCOS). References 1. A. Pásztor and D. Veitch, Active Probing using Packet Quartets, Proc. ACM SIGCOMM Internet Measurement Workshop (IMW-22), Marseille, (Nov 6-8, 22), pp C. Dovrolis, P. Ramanathan and D. Moore, Packet-Dispersion Techniques and a Capacity-Estimation Methodology, IEEE Transactions on Networking, Journal vol. 12, (Dec 24), pp D. R. Cox, Renewal Theory, Methuen & Co., London, (1962). 4. J.F. He and K. Sohraby, An Extended Combinatorial Analysis Framework for Discrete-Time Queueing Systems with General Sources, IEEE Transactions on Networking, Vol. 11, No. 1, pp , February (23). 5. L. Kleinrock, Queueing Systems, vol. 1, John Wiley and Sons, (1975). 6. L. Takács, Combinatorial Methods in the Theory of Stochastic Processes, John Wiley & Sons, Inc., New York, (1967). 7. M. Abramowitz and I. A. Stegun, Handbook of mathematical functions : with formulas, graphs, and mathematical tables, Dover Publications, New York, (1965). 8. O. Gurewitz, M. Sidi and I. Cidon, The ballot theorem Strikes Again: Packet Loss Process Distribution, IEEE Transactions on Information Theory, (2). 9. S. Keshav, Congestion Control in Computer Networks PhD Thesis, UC Berkeley TR-654, September (1991). 1. T. M. Apostol, Mathematical Analysis, Addison-Wesley, 2 nd edition, Massachusetts, (1981). 11. V. Ribeiro, R. Riedi, J. Navratil and L. Cottrell, pathchirp: Efficient Available Bandwidth Estimation for Network Paths,PAM 23, Passive and Active Measurement Workshop, La Jolla, California, (April 6-8, 23). 12. W. Matthews and L. Cottrell, Internet Performance Monitoring for the High Energy Nuclear and Particle Physics Community, in PAM 2, The First Passive and Active Measurement Workshop, program.htm, Hamilton, New Zealand, (Apr 2). SubintBP.tex; 16/9/25; 14:7; p.2