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Deriving delay characteristics from queue length statistics in discrete-time queues with multiple servers
ELSEVIER
Performance
Evaluation 24 (1996) 189-204
Deriving delay characteristics from queue length statistics in discrete-time queues with multiple servers Yijun Xiong, Herwig Bruneel *, Bart Steyaert Laboratory for Communications Engineering, University of Ghent, Sint-Pietersnieuwstraat 41, B-9000 Gent, Belgium Received 8 July 1992; revised March 1994
Abstract
In this paper, we investigate a discrete-time multiserver buffer system. Packets arrive in the system according to a general, possibly correlated, process, which is not further specified. The service times of the packets are of constant length. Explicit expressions are derived for the distribution, probability generating function, mean and variance of the packet delay, in terms of the distribution, probability generating function, mean and variance of the buffer contents. It is observed that knowledgeof the exact nature of the arrival process is not required in order to be able to derive these (general) relationships between the statistics of the delay and the occupancy. Keywords: Multiserver queue; Discrete time; Correlated Generalting function; ATM
arrivals; Fundamental
relationship;
Delay characteristics;
1. Introduction In the past several years, a lot of attention has been devoted to a wide variety of discrete-time queueing models. While both analytical and numerical results have been obtained on many occasions with respect to performance measures related to the buffer-contents distribution [l-11], for the case of multiple servers as well as for the single-server case, the derivation of delay characteristics has received much less attention in the past. In most cases, analytic results concerning the delay are limited to the mean value of the packet delay [3,4], which can be obtained by means of Little’s theorem, although other performance measures related to the delay, such as the variance and the tail distribution, are equally important for a wide range of applications, including system design in ATM-based B-ISDN networks. In a number of papers [3,4,7,8,12,13], delays are analyzed for single-server queueing
systems
* Corresponding
where
packets
arrive
according
author.
0166-5316/96/$15.00 0 1996 Elsevier Science B.V. All rights reserved SSDI 0166-5316(94)00031-E
to
an
uncorrelated
arrival
process.
In
190
I’. Xiong et al. /Performance Evaluation 24 (1996) 189-204
[6,14-161, the delays in various multiserver systems with uncorrelated arrival process are investigated. Finally, in [17-191, packet delays in single-server systems with various types of first-order Markovian correlated arrival processes are studied. In this paper, closed-form expressions are derived for the characteristics of the delay in discrete-time multiserver queues, in terms of the characteristics of the buffer (packet) contents, under the assumption of a very general, possibly correlated, arrival process (not necessarily first-order Markov). These relationships between buffer contents and delay are applicable for a wide variety of discrete-time queueing systems with multiple servers. They are, in fact, generalizations of analogous results for single-server queues, reported in [18,20]. In some general sense, the work reported in this paper is also related to various studies that have appeared in the literature on Little’s theorem and generalizations thereof (see, e.g., [21-23]), the main difference being that the present paper deals with a much more specific (restricted) class of queueing systems and (hence) obtains much stronger results (concerning the whole distribution of buffer contents and delay) than the aforementioned papers.
2. System description Throughout this paper, a discrete-time queueing system is studied under the following assumptions: - Time is divided into constant-length intervals, called slots, such that the service time of one packet (this is one unit of work) is equal to one slot. Services can only start at the beginning of the consecutive slots. - The buffer has an infinite storage capacity. - The queueing discipline is first-come-first-served (FCFS). - The system has c( > 1) servers, that are not subject to interruptions, i.e., the c packets (if present in the buffer) at the front of the queue at the beginning of a slot will leave the buffer at the end of this slot, since every packet is assumed to have a service time of exactly one slot. - Packets arrive in the buffer according to a general, possibly correlated, arrival process. The exact nature of the process (such as the nature of the correlation, the nature of the source(s) generating the packets) is irrelevant in the analysis following this section, as long as every packet has a service time of exactly one slot, and all packets are equally treated by the servers, i.e., there are no priorities. - We assume that the system reaches a steady state. This implies that the equilibrium condition, being the condition that the mean number of packet arrivals during an arbitrary slot is strictly less than c, is assumed to be satisfied. Let the random variable sk denote the system contents in the buffer at the beginning of slot k (this is the number of packets in the buffer at the beginning of slot k, including those that will be served during this slot), and let ek indicate the number of packets entering the buffer during slot k. Under the above assumptions, the following system equation can be established:
(1)
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191
where (.>‘a max(., 0). This equation expresses that the number of packets in the buffer at the beginning of a slot is equal to the sum of the number of packets in the buffer at the beginning of the previous slot and the number of packet arrivals during the previous slot, minus the number of packets that have left the system at the end of the previous slot (which is c, if at least c packets were present in the buffer at the beginning of the previous slot). In the following, we will denote by p(i, j> the steady-state joint probability p(i, j) P Pr[ s = i, e = j] 4 ,llm Pr[s, = i, ek = j],
(4
and by G(z, X) the associated joint probability generating function G(z, x)
b
5t i=O
z’dp(i,
j).
j=O
We have assumed here that the number of packet arrivals during a slot is limited by some integer N, where N can have any integer value between 1 and infinity. It is important to note here that at no point in the derivations it will be assumed that the couple (s, e) is a proper state description of our system, i.e., that this couple forms a two-dimensional Markov chain. This is a consequence of the fact that the knowledge of the exact nature of the process that generates the arriving packets will turn out to be irrelevant for the derivations that follow this section. From now on, we will merely assume that the characteristics of the random variable s, such as distribution, probability generating function, mean and variance, are known quantities.
3. Probability generating function of the packet delay We denote by d (with probability generating function D(z)) the random variable describing the delay of a randomly chosen (“tagged”) packet in the steady state. This is the number of slots between the end of the slot of arrival of that packet, and the end of the slot during which it leaves the system. When we concentrate on one arbitrary packet, we then find for its delay, under the assumptions made in the previous section, d=
(((s* - c)++f)
div C) + 1,
where (X div y) denotes the integer part of the fraction x/y. In this relation, s * denotes the system contents at the beginning of the slot of arrival of the tagged packet, and f is the number of packets that have arrived during the same slot as the tagged packet, but before it. In order to be able to derive the probability distribution of the packet delay from (41, we need an expression for the joint distribution of s * and f. A rigorous derivation of this distribution is beyond the scope of this paper (an approach as in [5] for a similar problem, for instance, could be used). Here we simply give a more intuitive “engineering-like” reasoning. In order to do so, we first write N
Pr[s* =i, f=j]
=
C l=j+l
Pr[s* =i, f=j,
e* cl],
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192
where e * is the number of packet arrivals during the arrival slot of the tagged packet. This equation can be written as Pr[s* =i, f=j]
=
5
=i,
Pr[f=j]s*
e * =I] Pr[s* =i,
e* =I].
l=j+l
Since f is the number of packets that have arrived in the system during the same slot as the tagged packet but before it, and the tagged packet has been picked randomly from a population of e * arriving packets, we find =i,
Pr[f=j]s*
e’ =I] =f,
O
Furthermore, since the tagged packet has been chosen randomly from all arriving packets, the joint probability Pr[s * = i, e * = Z] refers to the fraction ofpackets that arrive in a slot where there are Z packet arrivals and where the system contents at the beginning of the slot is equal to i. Since there are 1 arrivals in each such slot and the tagged packet could be any of these, we find
b(L I>
Pr[s* =i, e* =Z] = -
u
’
proportional to Zand proportional to p(i, Z), which can be regarded as the fraction ofslots with 1 packet arrivals and with a system contents of i packets at the start. The denominator (+ is the mean number of packet arrivals during an arbitrary slot, and ensures that the joint distribution of s * and e * is normalized. (Note that, due to the equilibrium condition, we must have a < c.) Combination of the three previous expressions leads to Pr[s* =i, f=j]
= 1 u
5
p(i, 1).
(5)
l=j+l
Defining H(z, X) as the joint probability generating function of the couple (s *, f), m N-l H(z,
x)
zS*xf]
%[
P
C
C
i=O j=O
Pr[s* =i, f=j],
zixi
we then obtain H(z,x)=l
i
ui=O
Nc1 j=fJ
f
z’xjp(i,
Changing the order of the summations N l-x’ H(Z,X)=J&iC1 --x p(i7 u
i=o
I).
l=j+l
I=1
and working out the sum over j yields z)*
With the definitions of the previous section, we obtain from this H(z, X) =
G(z, 1) - G(z, x) u(l-x)
*
A similar result was also derived in [17].
(6)
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We will now derive an expression for D(z) in terms of S(z), the probability generating function of the random variable s, being the system contents at the beginning of an arbitrary slot. From Eq. (41, we find D(z) =z C C zi Pr[(s” -c)++f=ic+j]. i=O j=O This can be transformed
into
C-l
D(zC)=zC
Cz-ji j=O
izn i=o
Pr[(s* -c)++f=n]S(ic+j-n),
n=O
where S(.) is the Kronecker-delta function, which is zero, unless its argument is zero, in which case it is 1. Let us now use the property that c-l
fJ6(ic-k),
Cdk=c
i= -_m
l=O
27ri
a g exp i
c
(7)
i’
(where i is the imaginary unit) that was also used in [6,12-151. This equation expresses that the sum in the left-hand side is zero unless k is a multiple of c, in which case a” is equal to 1 for all integer values of 1, and the sum becomes equal to c. The above expression for D(z”) can then be written as D(zc) = ~~$lz-jc~l i &n-j),n 1=0 n=O 1-O
Pr[(s” -c)++f=n].
This becomes, working out the sum for j and keeping in mind that arc = 1, for all integer values of 1, qzy
=
I” y
l-z-‘:_l c I=0 1 - (u’z)
i (a’~)~ Pr[(s” -c)++f=n]. n=O
If we denote by E[.] the expected value of the expression between the square brackets, we finally obtain qzc)
=
“‘c’ c
I=0
l -z-c_lE[(dlz)-)++f]. 1 - (a’z)
(8)
We now concentrate on the mean value in the right-hand side of the above expression. We find (+C’++f] =E[zS*-C+fj S* >c] Pr[s* >c] +E[zf)s* -CC] Pr[s*
-_E[zS*+fIs* +E[~f+~(s*
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194
Working out the right-hand side of this expression, we thus find c-l
s’+f] + C
Eb
i=O
\
N-l
C zj(z”-zi)
Pr[s* =i, f=j]
j=fJ
Using expression (5) and the definition of H(z, x), we obtain (S---C)++f]
E[z
zi(zc -z’)p(i,
F ”
\
i-0
j=O
I
or, using (61, and changing the order of the summations (s’-c)++f
Hz
G(z, 1) -G(z,
I--( -’ _ Z
z)
l-z
u
I)\,
l=j+l
over j and I,
c-1 N 1 -z’ -(zc +cc i=O/=l
-z’)p(i,
lez
1)
(9)
On the other hand, using the system equation (l), we get %+I] = E[ Z(“x-C)++ek].
E[z
With the definition of the steady-state joint probability generation function G(z, x), this can be translated (under the assumption that the system has reached its steady state) into the following equation: c-l N z”G( z, 1) = G( z, z) + C C z’(z’ -zi)p(i, I). (10) i=o I=0
Since G(z, 1) is equal to S(z), the probability generating function of the system contents at the beginning of an arbitrary slot, combining (9) and (10) yields (s. -c)+ +f
Eb
_
Z --c
l-z”
I--(
l_~S(z)
u
+
&~~‘(zc -z’)s(i) i-0
(11)
, 1
where s(i) is defined as Pr[ s = il. Combining (8) and (ll), we thus find an expression for D(z) in terms of the steady-state probability generating function S(z) of the system contents, and the c probabilities s(i), 0 < i Q c - 1:
l y
qzc)_
UC
I=0
1-z-c 1 -
(alz)-l
c-l
(a’z)Lz”
l-z’W4- jFo
i 1 -u’z
1
__a’z
49 . I
Defining
we finally obtain D(Z’)
=
h ~~~P,i(dZ)~')(
Po(U'Z)S(dZ)
- ‘c’,(dZ),,i)). i=O
(12)
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We note that, with the definition of P,(.), the second term in this expression for D(z) can be written as
i=O
I=0
Using the property (7), the sum over I in the right-hand obtain c-l ~~oPo((a~z)-l)
c
c-l
c-l
c-l Pi(a’z)s(i)
= c
c
c-i-l
c
2-j
j=O
i=O
i=O
side can be worked out. We then
c
rk+‘6(k
+
i -j)
k=O
where we have taken into account that -(c - 1) G k + i -j G c - 1, due to the upper and lower boundaries for i, j and k, which implies that k + i - j only is a multiple of c if k + i - j = 0. For each value of k and i, there is exactly one value of j for which k + i - j = 0. We thus obtain c-l
c-l
c-l
lzPo((a’z)el)
C
q(u’z)s(i)
= c C i=O
i=O
and this expression is independent q
c-i-l
c-l
C
s(i) = c C (c - i)s(i), i=O
k=O
of t. Equation (12) for D(z) then finally becomes
;~~Q(u’z)P(u’i)s(u’z) - ; y$(c - IBM
2”) = ;
l-0
(13)
where P(z) p PO(z), and Q(Z) p P(z-‘>. From this expression for D(z), further results will be derived. It is remarkable that the relationship between D(z) and S(z) does not explicitly depend on the arrival process. Once the distribution of the system contents has been determined, it is quite straightforward to calculate the distribution of the packet delay, as will be shown in Section 5. Before that,-we first examine the moments of the packet delay.
4. Moments of the packet delay In order to check the validity of the derivations in the previous section, we will verify that the normalization condition D(1) = 1 and Little’s theorem for the mean packet delay D’(1) are satisfied. Furthermore, we will derive an expression for the variance of the packet delay in terms of the statistics of the system contents. First of all, we establish some results that are useful in the following derivations. Using slightly non-standard notation, we define
P’(a’z) p
-&P(dz),P”(dZ) A $ydz).
We then find, with the definition of P(u’z), P(d)
=
c
j=O
c-l
c-l
c-l UQ,
P’(d)
=
C
j=O
ju’j,
P”(d)
= C j( j - 1)~‘~. j=O
(14ahc)
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In a similar way, taking the Oth, first and second derivatives with respect to z of Q(a’z> in t = 1, we obtain c-l
c-l Qb’)
=
p(J)
i~06
=
-
C
C-l
ja-lj,
p(d)
=
C
j( j + l)u+,
(15a,b,c)
j=O
j=O
where the derivatives for Q<.>have been defined in a similar way as those for PC.1 4.1. Norm&u
tion condition
Since P(u’) and Q are both equal to zero, unless I is equal to zero (in which case they are both equal to c), we find the following expression for D(1) by putting z = 1 into (13): - i ycl (C - i)s(i),
D(1) = AQ(l)P(l)
r=O
where we have used the property that the distribution of the system contents is normalized, i.e., S(1) = 1. Using (14a) and (15a) for 1= 0, we obtain D(1) = b c - ‘2 (c - i)s(i) I
i=O
This can be written as D(l)=:
‘cli
Pr[s=i]
+ ic
Pr[s=i]
i=c
i i=O
. 1
The quantity between parentheses is the mean number of packets that leave the system at the end of an arbitrary slot. It is well known that, for a system that has reached its steady state, this is equal to the mean number of packet arrivals during an arbitrary slot, which is u. We thus obtain D(1) = 1, as required. 4.2. Mean packet delay: Little’s theorem
In a similar way, by taking the first derivative of Eq. (13) with respect to z for z = 1, we obtain an expression for the mean packet delay D’(l): &‘(I)
= -$ [Q’(l)P(l)
+ Q(l)P’(l)
+
Q(l)I'(l)S'(l)].
This becomes, using (14a,b) and (15a,b) for I = 0 oCGD’(1) = -
c’(c - 1) 2
+
c”(c - 1) + C’S’(1). 2
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We thus obtain S’(1) D’(1) = u ’
(16)
which is in accordance with Little’s theorem. 4.3. Variance of the packet delay We will now derive an expression for the variance of the packet delay d. The variance of d can be expressed in terms of the first two derivatives of D(z) in z = 1 as var[d] =D”(l)
(17)
-D’(1)2+D’(l).
Consequently, we observe that we need an expression for the second derivative of D(2) in z = 1. Taking the second derivative of (13) in t = 1, we obtain c-l
(~c(c*D”(l) + c(c - l)D’(l))
= Q”(l)P(l)
+ Q(l)P”(l)
+ 2Q’(l)P(l)S’(l)
Using (14a,b,c) and (15a,b,c) for the appropriate = -&(c
- l)D’(l)
+
I=0
+ 2Q(l)P’(l)S’(l)
+ Q(l)P(l)S”(l).
values of I, this can be transformed
c”(c - 1)(2c - 1) a&“(l)
+ 2 c Q’(a’)P’(a’)S(a’)
3
c-l
c-l
into
c-l
- 2 /F. ,FojaP’j c
ka”‘S(a’)
+ c*S”(l).
k=O
With the property that c
CW
*
I I
c-l
a’-1
-1j
if 120,
’
c”(c - 1)’
j=O
9
4
if l=O,
the above expression for D”(1) becomes
c*- 1
D”(1) = 6ac
2 c-1 S(a’) - c (TC I=1 ]a’-lI*
+ -s”(1)
UC
- -D’(l). c-1 C
With the use of (16) and (17), we finally find vat-[ s] var[d] = -+--_ UC
c*- 1 6ac
2 c-1 c
UC
I=l
S(a’) (a’-
l)*
(18)
Thus, in order to calculate the variance of the packet delay d, we must evaluate the probability generating function S(z) of the system contents in the c solutions of the equation zc = 1, being (a’ IOG I G c - l] (and for higher-order moments, derivatives of S(z) will have to be evaluated in these points). Then the above equation allows the calculation of the variance of the packet delay in terms of the variance of the system contents.
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5. Distribution of the packet delay In this section, we will derive an expression for the distribution of the packet delay d, in terms of the distribution of the system contents s. Let us denote by d(n), IZ2 1, the steady-state probability that d is equal to n; this is the coefficient of 2” in the expressions (12) and (13) for D(z”). We prefer to depart from Eq. (12). This can be written as
l=O
i-0
k=O
j=O
Isolating in the first sum the terms for 0 < i < c - 1, and eliminating the sum over I using Eq. (71, we obtain
c-l +
c-l
c
c
i=l
s(j+i-k-C)tj+i-ks(i)
,
j=c-i
1
where we have used the property that in the second term, due to the lower and upper boundaries of i, j and k, we have 1 1; if n = 1 this will only be the case if k > j. In the second term, for all possible values of j and i, we always find exactly one value for 0 < k G c - 1 for which k = i + j - c. Taking into account the previous remarks, the Kronecker-delta functions disappear in the above formula when we work out the sum for i in the first and the sum for k in the second term, thus giving c-l d(z’)
=
m
c
c
k=O n=2
c-l
c-l
snc+k-j)+z”z
c
j=O
j=O
We then obtain, after some manipulations expression for D( 2’)
c-l k=j
with the summation
c-l d(z”)
=
2
n=2
c-l
i=l
j=c-i
c boundaries
s(i). in the previous
2(c - 1)
(I+1)s((n-l)c+Z+l)+
c
znc
c-l
xs(c+k-j)+z”x
z lzc (
i l=O
c
(2c-l-l)s((n-l)c+Z+l)
l=c
I
c-l +zc
C
(is(i) + (c - i)s(i + c)).
i=O
Keeping in mind that d(n) is the coefficient of znc in the power expansion for D( z’), after some algebraic manipulations, we finally find the following expression for d(n): c-l ad(n)
=
C
i=O
{is((n
-
1)c + i) + (c - i)s(nc + i)},
n 2 1.
(1%
Note that this formula shows that d(n) can be obtained as a weighted sum of s(nc) and the system-contents probabilities in the “neighborhood” of nc, i.e., s(nc + l), s(nc + 2), . . . , s(nc +
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199
(c - l)), where the weight for s(nc + k) decreases linearly with k. From this formula, we can easily derive an expression for the probability that the packet delay exceeds a given integer value D, D 2 1: c-l CT Pr[d>D]
=c Pr[sacD]
+ CZs((D-l)c+Z). I=1
(20)
Once the distribution of the system contents s has been established, Eqs. (19) and (20) allow the calculation of the distribution of the packet delay, and the probability that the packet delay d exceeds a given integer value respectively. The calculations required to derive these two quantities are remarkably simple. Again, we note that, once the distribution (and/or the probability generating function) of the system contents has been derived, the knowledge of the exact nature of the arrival process is no longer needed in order to be able to derive the performance characteristics of the packet delay. That is, the behavior of the system contents as well as the delay may be strongly dependent on the exact nature of the arrival process, but the transformation from system contents to delay is not.
6. Application to ATM switches We have compared the results derived in this paper with simulation results, for two specific discrete-time queueing systems that can be used to model ATM switches. ATM switching networks consist of a matrix of elementary ATM switches that provide the routing of ATM cells (the equivalent of a packet in ATM) sent through the network. In the following examples, we focus attention on one specific ATM switch (Fig. 1). Cells enter the switch via one of its input links, and are then routed to one of the output links, where they are temporarily buffered in a designated output queue (O.Q.) to await transmission of the cells that arrived before them in this particular output queue and that are not yet sent. The results of this paper are applicable to assess the delay performance in such a multiple-server output queue, under the assumption that each output queue (corresponding with one destination), has a constant number of transmission lines (denoted by the integer c 2 1) via which the cells are transmitted. As shown in Fig. 1, in the following examples, N denotes the number of input and output links of a switch. Cell arrivals on different input links are assumed to be independent (since they originate from different sources), but the arrival process on one particular inlet can be either independent or correlated from slot to slot. Furthermore, it is assumed that the routing
Fig. 1. ATM switch with multiserver output queues.
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-c g
Evaluation
24 (1996) 189-204
1
1
IOOE-02 -4 t
1
Fig. 2. Delay distribution
in an ATM switch with uncorrelated
arrivals on the inlets: N = 16, c = 1,2,4, p = 0.8.
of cells from the input links to the N/c output queues is performed in a uniform and independent way, i.e., each arriving cell is routed to any given output queue with probability c/N (since there are N/c possible destinations), independently of the routing of other cells. Such a model is reasonable if the cell streams on the inlets of the switch are composed of many independent multiplexed cell streams originating from users whose peak cell rate is very low in comparison with the link rate. 6.1. ATM switch with uncorrelated arrival processes on the inlets Let us first assume that the arrival process on each inlet is uncorrelated (in other words, the numbers of cell arrivals during consecutive slots are i.i.d.), and that with probability p (the load on each inlet) there will be one cell arrival during a slot on a tagged inlet, and with probability 1 -p there will be no cell arrival, that is, the number of cell arrivals during each slot on an inlet is Bernoulli distributed with parameter p. As we already mentioned, arriving cells are then routed in a uniform and independent way to the output queue assigned to their destination, which implies that the total offered load to one output queue is (T=pc. In Fig. 2, we have plotted the delay distribution d(n) versus n, for N = 16, c = 1, 2, 4, and p = 0.8. These results have been obtained through computer simulation; the solid lines connect the simulated data obtained for the delay distribution, while the data represented by the dots
I’. Xiong et al. /Performance Evaluation 24 (1996) 189-204
201
have been calculated using Eq. (19), whereby the buffer-contents distribution s(n) was also obtained through simulation. Comparison of these results gives a perfect match, which confirms the validity of the analysis presented in this paper. Note that for the model considered here, explicit expressions for the probability generating functions of the buffer contents and the packet delay are available in [14] and/or [20]. 6.2. ATM switch with correlated arrival processes on the inlets We now consider the case where the cell arrival process on each inlet of the switch is first-order Markov (note that due to the routing of arriving cells to their output queues, the cell arrival process in an output queue is no longer first-order Markov). If we define an active period on an input link as a number of consecutive slots during which a cell arrival occurs, and, similarly, a passive period as a number of consecutive slots during which there is no cell arrival on this input link, then the first-order Markov assumption means that the lengths of both active and passive periods are geometrically distributed random variables with parameters (Yand p respectively, i.e., Pr[an active period contains II slots] = (1 - ~y)~~-l,
n 2 1,
Pr[a passive period contains n slots] = (1 - j3)pnw1,
it 2 1.
I
OOE-01
Fig. 3. Delay distribution in an ATM switch with correlated c = 1,2,4, p = 0.8, K = 5.
(first-order
Markov) arrivals on the inlets: N = 16,
202
Y. Xiong et al. /Performance Evaluation 24 (1996) 189-204
Furthermore, lengths of consecutive active (or passive) periods are i.i.d. Note that the case of an uncorrelated Bernoulli arrival law on each input link of the switch corresponds to (Y+ p = 1. The couple (a, p) fully describes the arrival process on each of the inlets. Nevertheless, we prefer to use a pair of parameters that allow a more intuitive interpretation. First of all, defining p as the average load on each of the inlets of the switch, (i.e., (T=pc>, we find that p = (1 - p)/(2 - LY - p). Furthermore, we define K as the ratio of the mean length of an active (or passive) period in the correlated case versus the mean length of an active (or passive) period in the uncorrelated case with same (mean) load p. This implies that K p (1 -a)/(1 - a> = p/(1 - p). Thus, the case of an uncorrelated Bernoulli arrival law on each individual inlet corresponds to K = 1, and it is clear that the couple (p, K) also fully describes the cell arrival process on the inlets. In Fig. 3, we have plotted the delay distribution d(n) versus n, for N = 16, c = 1, 2, 4, p=O.8 and K=5. Th ese results have been obtained through simulation; again, the solid lines connect the simulated data obtained for the delay distribution, while the data represented by the dots have been calculated using Eq. (191, whereby the buffer-contents distribution was also obtained through simulation. Also for this case, comparison of these results gives a perfect match, and confirms the analysis presented in this paper, as we expected. Note that for the single-server case (c = 1) in the model considered here, the performance characteristics of the buffer contents were extensively treated in [25]. 7. Conclusions In this paper, we have established a relationship between the packet delay and the buffer contents in a discrete-time multiserver queueing system without server interruptions, where the service times of the packets are equal to one slot, and where the arrival process of the packets in the buffer is a general (not necessarily Markovian) correlated process. Specifically, explicit expressions have been derived for the distribution, the probability generating function, the mean (Little’s theorem) and the variance of the packet delay, in terms of the distribution, the probability generating function, the mean and the variance of the buffer contents. These results have been illustrated for two discrete-time queueing systems that can be used to model ATM switching elements. One of the main observations is that knowledge of the exact nature of the arrival process is not required in order to be able to derive these general relationships between the statistics of the delay and the buffer occupancy. Acknowledgement
The second author wishes to acknowledge Scientific Research (N.F.W.O.).
the support of the Belgian National Fund for
References [l] W.W. Chu and A.G. Konheim, On the analysis and modeling of a class of computer communication IEEE Trans. Comm. 20 (1972) 645-660.
systems,
Y. Xiong et al. /Performance Evaluation 24 (1996) 189-204
203
[2] N.D. Georganas, Buffer behavior with Poisson arrivals and bulk geometric service, ZEEE Trans. Comm. 24 (1976) 938-940. [3] S.S. Lam, Delay analysis of a time division multiple access (TDMA) channel, IEEE Trans. Comm. 25 (1977) 1489-1494. [4] I. Rubin, Message delays in FDMA and TDMA communication channels, IEEE Trans. Comm. 27 (1979) 769-777. [5] H. Bruneel, A general treatment of discrete-time buffers with one randomly interrupted output line, Eur. .Z. Oper. Res. 27 (1986) 67-81. [6] H. Bruneel, Message delay in TDMA channels with contiguous output, IEEE Trans. Comm. 34 (1986) 681-684. [7] M.J. Karol, M.G. Hluchyj and S.P. Morgan, Input versus output queueing on a space-division packet switch, IEEE Trans. Comm. 35 (1987) 1347-1356. [8] A. Gravey, J.-R. Louvion and P. Boyer, On the Gee/D/l and Gee/D/l/n queues, Perf Euul. 11 (1990) 117-125. 191 E. Desmet and G.H. Petit, Performance analysis of the discrete-time multiserver queueing system GedN)/D/c/K, RACE Workshop, Munich, 3-4 July 1990. [lo] G.H. Petit and E. Desmet, Performance evaluation of shared multiserver output queue switches used in ATM, 7th ZTC Specialist Seminur, Morristown, N.J., 1990. [ll] S. Li, A general solution technique for discrete queueing analysis of multimedia traffic on ATM, IEEE Trans. Comm. 39 (1991) 1115-1132. [12] K.-T. Ko and B.R. Davis, Delay analysis for a TDMA channel with contiguous output and Poisson message arrival, IEEE Trans. Comm. 32 (1984) 707-709. [13] B. Steyaert and H. Bruneel, A general analysis of the packet delay in TDMA channels with contiguous-slot assignment, Proc. ZCC’9Z, Denver, Cola., June 1991, Vol. 3, pp. 1539-1543. [14] H. Bruneel, B. Steyaert, E. Desmet and G.H. Petit, An analytic technique for the derivation of the delay performance of ATM switches with multiserver output queues, Znt. J. Digit. & Analog Comm. Syst. 5 (1992) 193-201. [15] E. Desmet, B. Steyaert, H. Bruneel and G.H. Petit, Tail distributions of queue length and delay in discrete-time multiserver queueing models, applicable in ATM networks, Proc. ZTC’13 Copenhagen, June 1991, Vol. Queueing, Performance and Control in ATM, pp. l-6. [16] K. Laevens and H. Bruneel, Delay analysis for discrete-time queueing systems with multiple randomly interrupted servers, Eur. J. Oper. Res. 85 (1995) 161-177. [17] H. Bruneel, Packet delay and queue length for statistical multiplexers with low-speed access lines, Comput. Networks ZSDN Syst. 25 (1993) 1267-1277. [18] Y. Xiong and H. Bruneel, Buffer contents and delay for statistical multiplexers with fixed-length packet-train arrivals, Perf Euul. 17 (1993) 31-42. [19] B. Steyaert and H. Bruneel, An effective algorithm to calculate the distribution of the buffer contents and the packet delay in a multiplexer with bursty sources, Proc. GLOBECOMPI, Phoenix, AZ., December 1991, Vol. 1, pp. 471-475. [20] H. Bruneel and B.G. Kim, Discrete-time Models for Communication Systems Including ATM, Khtwer Academic Publishers, Boston (1993). [21] S.L. Brumelle, A generalization of L = AW to moments of queue length and waiting times, Oper. Res. 20 (1972) 1127-1136. [22] D.P. Heyman and S. Stidham, The relation between customer and time averages in queues, Oper. Res. 28 (1980) 983-994. [23] P.W. Glynn and W. Whitt, Extensions of the queueing relations L = AW and H = AG, Oper. Res. 37 (1989) 634-644. [24] I. Rubin, Access control disciplines for multi-access communication channels: reservation and ‘IDMA schemes, IEEE Trans. Inform. Theory 25 (1979) 516-536. [25] B. Steyaert and H. Bruneel, Analytical study of the buffer contents in an ATM switch with Markovian arrivals, Proc. ZSTN-92, Beijing, China, September 1992, pp. 9-12.
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Yiiun Xiong was born in Wuhan, P.R. China, in 1963. He received the B.S. and M.S. degrees from Shanghai Jiao-Tong University, Shanghai, P.R. China, in 1984 and 1987, respectively, and the Ph.D. degree from the University of Ghent, Gent, Belgium, in 1994, all in electrical engineering. From January 1987 to Jyly 1989 he was a Ph.D. student at the Southeast University, Nanjing, P.R. China. From August 1989 to August 1992, he was with the Research Center of Alcatel Bell Telephone in Antwerp, Belgium. Since September 1992, he has been a scientific researcher at the Laboratory for Communications Engineering, University of Ghent, Belgium. His current research interests include the modeling and performance analysis of communications systems, traffic control and management in high-speed networks, queueing theory and simulation.
Herwig Bruneel was born in Zottegem, Belgium, in 1954. He received the MS. degree in electrical engineering, the degree of Licentiate in computer science, and the Ph.D. degree in computer science in 1978, 1979 and 1984, respectively, ail from the University of Ghent, Ghent, Belgium. Since 1979, he has been working as a researcher for the Belgian National Fund for Scientific Research (N.F.W.O.) at the University of Ghent. He is also a PRofessor in the Faculty of Applied Sciences at the same university. His main researc interests include stochastic modeling of digital communications systems, , discrete-time queueing theory, and the study of ARQ protocols. He has published more than 1 70 papers on these subjects and is coauthor of the book H. Bruneel and B.G. Kim, “DiscreteTime Models for Communication Systems Including ATM” (Kluwer Academic Publishers, Boston, 1993).
Bart Steyart was born in Roeselare, Belgium, in 1964. He received the degrees of Licentiate in Physics and Licentiate in Computer Science in 1987 and 1989, respectively, all from the University of Ghent, Ghent, Belgium. Since 1990, he has been working as a Ph.D. student at the Laboratory for Communications Engineering, University of Ghent. His main research interests include performance evaluation of discrete-time queueing models.
Performance
Evaluation 24 (1996) 189-204
Deriving delay characteristics from queue length statistics in discrete-time queues with multiple servers Yijun Xiong, Herwig Bruneel *, Bart Steyaert Laboratory for Communications Engineering, University of Ghent, Sint-Pietersnieuwstraat 41, B-9000 Gent, Belgium Received 8 July 1992; revised March 1994
Abstract
In this paper, we investigate a discrete-time multiserver buffer system. Packets arrive in the system according to a general, possibly correlated, process, which is not further specified. The service times of the packets are of constant length. Explicit expressions are derived for the distribution, probability generating function, mean and variance of the packet delay, in terms of the distribution, probability generating function, mean and variance of the buffer contents. It is observed that knowledgeof the exact nature of the arrival process is not required in order to be able to derive these (general) relationships between the statistics of the delay and the occupancy. Keywords: Multiserver queue; Discrete time; Correlated Generalting function; ATM
arrivals; Fundamental
relationship;
Delay characteristics;
1. Introduction In the past several years, a lot of attention has been devoted to a wide variety of discrete-time queueing models. While both analytical and numerical results have been obtained on many occasions with respect to performance measures related to the buffer-contents distribution [l-11], for the case of multiple servers as well as for the single-server case, the derivation of delay characteristics has received much less attention in the past. In most cases, analytic results concerning the delay are limited to the mean value of the packet delay [3,4], which can be obtained by means of Little’s theorem, although other performance measures related to the delay, such as the variance and the tail distribution, are equally important for a wide range of applications, including system design in ATM-based B-ISDN networks. In a number of papers [3,4,7,8,12,13], delays are analyzed for single-server queueing
systems
* Corresponding
where
packets
arrive
according
author.
0166-5316/96/$15.00 0 1996 Elsevier Science B.V. All rights reserved SSDI 0166-5316(94)00031-E
to
an
uncorrelated
arrival
process.
In
190
I’. Xiong et al. /Performance Evaluation 24 (1996) 189-204
[6,14-161, the delays in various multiserver systems with uncorrelated arrival process are investigated. Finally, in [17-191, packet delays in single-server systems with various types of first-order Markovian correlated arrival processes are studied. In this paper, closed-form expressions are derived for the characteristics of the delay in discrete-time multiserver queues, in terms of the characteristics of the buffer (packet) contents, under the assumption of a very general, possibly correlated, arrival process (not necessarily first-order Markov). These relationships between buffer contents and delay are applicable for a wide variety of discrete-time queueing systems with multiple servers. They are, in fact, generalizations of analogous results for single-server queues, reported in [18,20]. In some general sense, the work reported in this paper is also related to various studies that have appeared in the literature on Little’s theorem and generalizations thereof (see, e.g., [21-23]), the main difference being that the present paper deals with a much more specific (restricted) class of queueing systems and (hence) obtains much stronger results (concerning the whole distribution of buffer contents and delay) than the aforementioned papers.
2. System description Throughout this paper, a discrete-time queueing system is studied under the following assumptions: - Time is divided into constant-length intervals, called slots, such that the service time of one packet (this is one unit of work) is equal to one slot. Services can only start at the beginning of the consecutive slots. - The buffer has an infinite storage capacity. - The queueing discipline is first-come-first-served (FCFS). - The system has c( > 1) servers, that are not subject to interruptions, i.e., the c packets (if present in the buffer) at the front of the queue at the beginning of a slot will leave the buffer at the end of this slot, since every packet is assumed to have a service time of exactly one slot. - Packets arrive in the buffer according to a general, possibly correlated, arrival process. The exact nature of the process (such as the nature of the correlation, the nature of the source(s) generating the packets) is irrelevant in the analysis following this section, as long as every packet has a service time of exactly one slot, and all packets are equally treated by the servers, i.e., there are no priorities. - We assume that the system reaches a steady state. This implies that the equilibrium condition, being the condition that the mean number of packet arrivals during an arbitrary slot is strictly less than c, is assumed to be satisfied. Let the random variable sk denote the system contents in the buffer at the beginning of slot k (this is the number of packets in the buffer at the beginning of slot k, including those that will be served during this slot), and let ek indicate the number of packets entering the buffer during slot k. Under the above assumptions, the following system equation can be established:
(1)
I’. Xiong et al. /Performance Evaluation 24 (1996) 189-204
191
where (.>‘a max(., 0). This equation expresses that the number of packets in the buffer at the beginning of a slot is equal to the sum of the number of packets in the buffer at the beginning of the previous slot and the number of packet arrivals during the previous slot, minus the number of packets that have left the system at the end of the previous slot (which is c, if at least c packets were present in the buffer at the beginning of the previous slot). In the following, we will denote by p(i, j> the steady-state joint probability p(i, j) P Pr[ s = i, e = j] 4 ,llm Pr[s, = i, ek = j],
(4
and by G(z, X) the associated joint probability generating function G(z, x)
b
5t i=O
z’dp(i,
j).
j=O
We have assumed here that the number of packet arrivals during a slot is limited by some integer N, where N can have any integer value between 1 and infinity. It is important to note here that at no point in the derivations it will be assumed that the couple (s, e) is a proper state description of our system, i.e., that this couple forms a two-dimensional Markov chain. This is a consequence of the fact that the knowledge of the exact nature of the process that generates the arriving packets will turn out to be irrelevant for the derivations that follow this section. From now on, we will merely assume that the characteristics of the random variable s, such as distribution, probability generating function, mean and variance, are known quantities.
3. Probability generating function of the packet delay We denote by d (with probability generating function D(z)) the random variable describing the delay of a randomly chosen (“tagged”) packet in the steady state. This is the number of slots between the end of the slot of arrival of that packet, and the end of the slot during which it leaves the system. When we concentrate on one arbitrary packet, we then find for its delay, under the assumptions made in the previous section, d=
(((s* - c)++f)
div C) + 1,
where (X div y) denotes the integer part of the fraction x/y. In this relation, s * denotes the system contents at the beginning of the slot of arrival of the tagged packet, and f is the number of packets that have arrived during the same slot as the tagged packet, but before it. In order to be able to derive the probability distribution of the packet delay from (41, we need an expression for the joint distribution of s * and f. A rigorous derivation of this distribution is beyond the scope of this paper (an approach as in [5] for a similar problem, for instance, could be used). Here we simply give a more intuitive “engineering-like” reasoning. In order to do so, we first write N
Pr[s* =i, f=j]
=
C l=j+l
Pr[s* =i, f=j,
e* cl],
Y: Xiong et al. /Performance Evaluation 24 (1996) 189-204
192
where e * is the number of packet arrivals during the arrival slot of the tagged packet. This equation can be written as Pr[s* =i, f=j]
=
5
=i,
Pr[f=j]s*
e * =I] Pr[s* =i,
e* =I].
l=j+l
Since f is the number of packets that have arrived in the system during the same slot as the tagged packet but before it, and the tagged packet has been picked randomly from a population of e * arriving packets, we find =i,
Pr[f=j]s*
e’ =I] =f,
O
Furthermore, since the tagged packet has been chosen randomly from all arriving packets, the joint probability Pr[s * = i, e * = Z] refers to the fraction ofpackets that arrive in a slot where there are Z packet arrivals and where the system contents at the beginning of the slot is equal to i. Since there are 1 arrivals in each such slot and the tagged packet could be any of these, we find
b(L I>
Pr[s* =i, e* =Z] = -
u
’
proportional to Zand proportional to p(i, Z), which can be regarded as the fraction ofslots with 1 packet arrivals and with a system contents of i packets at the start. The denominator (+ is the mean number of packet arrivals during an arbitrary slot, and ensures that the joint distribution of s * and e * is normalized. (Note that, due to the equilibrium condition, we must have a < c.) Combination of the three previous expressions leads to Pr[s* =i, f=j]
= 1 u
5
p(i, 1).
(5)
l=j+l
Defining H(z, X) as the joint probability generating function of the couple (s *, f), m N-l H(z,
x)
zS*xf]
%[
P
C
C
i=O j=O
Pr[s* =i, f=j],
zixi
we then obtain H(z,x)=l
i
ui=O
Nc1 j=fJ
f
z’xjp(i,
Changing the order of the summations N l-x’ H(Z,X)=J&iC1 --x p(i7 u
i=o
I).
l=j+l
I=1
and working out the sum over j yields z)*
With the definitions of the previous section, we obtain from this H(z, X) =
G(z, 1) - G(z, x) u(l-x)
*
A similar result was also derived in [17].
(6)
Y. Xiong et al. /Performance
Evaluation 24 (1996) 189-204
193
We will now derive an expression for D(z) in terms of S(z), the probability generating function of the random variable s, being the system contents at the beginning of an arbitrary slot. From Eq. (41, we find D(z) =z C C zi Pr[(s” -c)++f=ic+j]. i=O j=O This can be transformed
into
C-l
D(zC)=zC
Cz-ji j=O
izn i=o
Pr[(s* -c)++f=n]S(ic+j-n),
n=O
where S(.) is the Kronecker-delta function, which is zero, unless its argument is zero, in which case it is 1. Let us now use the property that c-l
fJ6(ic-k),
Cdk=c
i= -_m
l=O
27ri
a g exp i
c
(7)
i’
(where i is the imaginary unit) that was also used in [6,12-151. This equation expresses that the sum in the left-hand side is zero unless k is a multiple of c, in which case a” is equal to 1 for all integer values of 1, and the sum becomes equal to c. The above expression for D(z”) can then be written as D(zc) = ~~$lz-jc~l i &n-j),n 1=0 n=O 1-O
Pr[(s” -c)++f=n].
This becomes, working out the sum for j and keeping in mind that arc = 1, for all integer values of 1, qzy
=
I” y
l-z-‘:_l c I=0 1 - (u’z)
i (a’~)~ Pr[(s” -c)++f=n]. n=O
If we denote by E[.] the expected value of the expression between the square brackets, we finally obtain qzc)
=
“‘c’ c
I=0
l -z-c_lE[(dlz)-)++f]. 1 - (a’z)
(8)
We now concentrate on the mean value in the right-hand side of the above expression. We find (+C’++f] =E[zS*-C+fj S* >c] Pr[s* >c] +E[zf)s* -CC] Pr[s*
-_E[zS*+fIs* +E[~f+~(s*
Y. Xiong et al. /Performance Evaluation 24 (1996) 189-204
194
Working out the right-hand side of this expression, we thus find c-l
s’+f] + C
Eb
i=O
\
N-l
C zj(z”-zi)
Pr[s* =i, f=j]
j=fJ
Using expression (5) and the definition of H(z, x), we obtain (S---C)++f]
E[z
zi(zc -z’)p(i,
F ”
\
i-0
j=O
I
or, using (61, and changing the order of the summations (s’-c)++f
Hz
G(z, 1) -G(z,
I--( -’ _ Z
z)
l-z
u
I)\,
l=j+l
over j and I,
c-1 N 1 -z’ -(zc +cc i=O/=l
-z’)p(i,
lez
1)
(9)
On the other hand, using the system equation (l), we get %+I] = E[ Z(“x-C)++ek].
E[z
With the definition of the steady-state joint probability generation function G(z, x), this can be translated (under the assumption that the system has reached its steady state) into the following equation: c-l N z”G( z, 1) = G( z, z) + C C z’(z’ -zi)p(i, I). (10) i=o I=0
Since G(z, 1) is equal to S(z), the probability generating function of the system contents at the beginning of an arbitrary slot, combining (9) and (10) yields (s. -c)+ +f
Eb
_
Z --c
l-z”
I--(
l_~S(z)
u
+
&~~‘(zc -z’)s(i) i-0
(11)
, 1
where s(i) is defined as Pr[ s = il. Combining (8) and (ll), we thus find an expression for D(z) in terms of the steady-state probability generating function S(z) of the system contents, and the c probabilities s(i), 0 < i Q c - 1:
l y
qzc)_
UC
I=0
1-z-c 1 -
(alz)-l
c-l
(a’z)Lz”
l-z’W4- jFo
i 1 -u’z
1
__a’z
49 . I
Defining
we finally obtain D(Z’)
=
h ~~~P,i(dZ)~')(
Po(U'Z)S(dZ)
- ‘c’,(dZ),,i)). i=O
(12)
Y. Xiong et al. /Performance
195
Evaluation 24 (1996) 189-204
We note that, with the definition of P,(.), the second term in this expression for D(z) can be written as
i=O
I=0
Using the property (7), the sum over I in the right-hand obtain c-l ~~oPo((a~z)-l)
c
c-l
c-l
c-l Pi(a’z)s(i)
= c
c
c-i-l
c
2-j
j=O
i=O
i=O
side can be worked out. We then
c
rk+‘6(k
+
i -j)
k=O
where we have taken into account that -(c - 1) G k + i -j G c - 1, due to the upper and lower boundaries for i, j and k, which implies that k + i - j only is a multiple of c if k + i - j = 0. For each value of k and i, there is exactly one value of j for which k + i - j = 0. We thus obtain c-l
c-l
c-l
lzPo((a’z)el)
C
q(u’z)s(i)
= c C i=O
i=O
and this expression is independent q
c-i-l
c-l
C
s(i) = c C (c - i)s(i), i=O
k=O
of t. Equation (12) for D(z) then finally becomes
;~~Q(u’z)P(u’i)s(u’z) - ; y$(c - IBM
2”) = ;
l-0
(13)
where P(z) p PO(z), and Q(Z) p P(z-‘>. From this expression for D(z), further results will be derived. It is remarkable that the relationship between D(z) and S(z) does not explicitly depend on the arrival process. Once the distribution of the system contents has been determined, it is quite straightforward to calculate the distribution of the packet delay, as will be shown in Section 5. Before that,-we first examine the moments of the packet delay.
4. Moments of the packet delay In order to check the validity of the derivations in the previous section, we will verify that the normalization condition D(1) = 1 and Little’s theorem for the mean packet delay D’(1) are satisfied. Furthermore, we will derive an expression for the variance of the packet delay in terms of the statistics of the system contents. First of all, we establish some results that are useful in the following derivations. Using slightly non-standard notation, we define
P’(a’z) p
-&P(dz),P”(dZ) A $ydz).
We then find, with the definition of P(u’z), P(d)
=
c
j=O
c-l
c-l
c-l UQ,
P’(d)
=
C
j=O
ju’j,
P”(d)
= C j( j - 1)~‘~. j=O
(14ahc)
Y. Xiong et al. /Performance
196
Evaluation 24 (1996) 189-204
In a similar way, taking the Oth, first and second derivatives with respect to z of Q(a’z> in t = 1, we obtain c-l
c-l Qb’)
=
p(J)
i~06
=
-
C
C-l
ja-lj,
p(d)
=
C
j( j + l)u+,
(15a,b,c)
j=O
j=O
where the derivatives for Q<.>have been defined in a similar way as those for PC.1 4.1. Norm&u
tion condition
Since P(u’) and Q
D(1) = AQ(l)P(l)
r=O
where we have used the property that the distribution of the system contents is normalized, i.e., S(1) = 1. Using (14a) and (15a) for 1= 0, we obtain D(1) = b c - ‘2 (c - i)s(i) I
i=O
This can be written as D(l)=:
‘cli
Pr[s=i]
+ ic
Pr[s=i]
i=c
i i=O
. 1
The quantity between parentheses is the mean number of packets that leave the system at the end of an arbitrary slot. It is well known that, for a system that has reached its steady state, this is equal to the mean number of packet arrivals during an arbitrary slot, which is u. We thus obtain D(1) = 1, as required. 4.2. Mean packet delay: Little’s theorem
In a similar way, by taking the first derivative of Eq. (13) with respect to z for z = 1, we obtain an expression for the mean packet delay D’(l): &‘(I)
= -$ [Q’(l)P(l)
+ Q(l)P’(l)
+
Q(l)I'(l)S'(l)].
This becomes, using (14a,b) and (15a,b) for I = 0 oCGD’(1) = -
c’(c - 1) 2
+
c”(c - 1) + C’S’(1). 2
Y. Xiong et al./Performance
197
Evaluation 24 (1996) 189-204
We thus obtain S’(1) D’(1) = u ’
(16)
which is in accordance with Little’s theorem. 4.3. Variance of the packet delay We will now derive an expression for the variance of the packet delay d. The variance of d can be expressed in terms of the first two derivatives of D(z) in z = 1 as var[d] =D”(l)
(17)
-D’(1)2+D’(l).
Consequently, we observe that we need an expression for the second derivative of D(2) in z = 1. Taking the second derivative of (13) in t = 1, we obtain c-l
(~c(c*D”(l) + c(c - l)D’(l))
= Q”(l)P(l)
+ Q(l)P”(l)
+ 2Q’(l)P(l)S’(l)
Using (14a,b,c) and (15a,b,c) for the appropriate = -&(c
- l)D’(l)
+
I=0
+ 2Q(l)P’(l)S’(l)
+ Q(l)P(l)S”(l).
values of I, this can be transformed
c”(c - 1)(2c - 1) a&“(l)
+ 2 c Q’(a’)P’(a’)S(a’)
3
c-l
c-l
into
c-l
- 2 /F. ,FojaP’j c
ka”‘S(a’)
+ c*S”(l).
k=O
With the property that c
CW
*
I I
c-l
a’-1
-1j
if 120,
’
c”(c - 1)’
j=O
9
4
if l=O,
the above expression for D”(1) becomes
c*- 1
D”(1) = 6ac
2 c-1 S(a’) - c (TC I=1 ]a’-lI*
+ -s”(1)
UC
- -D’(l). c-1 C
With the use of (16) and (17), we finally find vat-[ s] var[d] = -+--_ UC
c*- 1 6ac
2 c-1 c
UC
I=l
S(a’) (a’-
l)*
(18)
Thus, in order to calculate the variance of the packet delay d, we must evaluate the probability generating function S(z) of the system contents in the c solutions of the equation zc = 1, being (a’ IOG I G c - l] (and for higher-order moments, derivatives of S(z) will have to be evaluated in these points). Then the above equation allows the calculation of the variance of the packet delay in terms of the variance of the system contents.
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5. Distribution of the packet delay In this section, we will derive an expression for the distribution of the packet delay d, in terms of the distribution of the system contents s. Let us denote by d(n), IZ2 1, the steady-state probability that d is equal to n; this is the coefficient of 2” in the expressions (12) and (13) for D(z”). We prefer to depart from Eq. (12). This can be written as
l=O
i-0
k=O
j=O
Isolating in the first sum the terms for 0 < i < c - 1, and eliminating the sum over I using Eq. (71, we obtain
c-l +
c-l
c
c
i=l
s(j+i-k-C)tj+i-ks(i)
,
j=c-i
1
where we have used the property that in the second term, due to the lower and upper boundaries of i, j and k, we have 1
=
m
c
c
k=O n=2
c-l
c-l
snc+k-j)+z”z
c
j=O
j=O
We then obtain, after some manipulations expression for D( 2’)
c-l k=j
with the summation
c-l d(z”)
=
2
n=2
c-l
i=l
j=c-i
c boundaries
s(i). in the previous
2(c - 1)
(I+1)s((n-l)c+Z+l)+
c
znc
c-l
xs(c+k-j)+z”x
z lzc (
i l=O
c
(2c-l-l)s((n-l)c+Z+l)
l=c
I
c-l +zc
C
(is(i) + (c - i)s(i + c)).
i=O
Keeping in mind that d(n) is the coefficient of znc in the power expansion for D( z’), after some algebraic manipulations, we finally find the following expression for d(n): c-l ad(n)
=
C
i=O
{is((n
-
1)c + i) + (c - i)s(nc + i)},
n 2 1.
(1%
Note that this formula shows that d(n) can be obtained as a weighted sum of s(nc) and the system-contents probabilities in the “neighborhood” of nc, i.e., s(nc + l), s(nc + 2), . . . , s(nc +
Y. Xiong et al./Performance Evaluation 24 (1996) 189-204
199
(c - l)), where the weight for s(nc + k) decreases linearly with k. From this formula, we can easily derive an expression for the probability that the packet delay exceeds a given integer value D, D 2 1: c-l CT Pr[d>D]
=c Pr[sacD]
+ CZs((D-l)c+Z). I=1
(20)
Once the distribution of the system contents s has been established, Eqs. (19) and (20) allow the calculation of the distribution of the packet delay, and the probability that the packet delay d exceeds a given integer value respectively. The calculations required to derive these two quantities are remarkably simple. Again, we note that, once the distribution (and/or the probability generating function) of the system contents has been derived, the knowledge of the exact nature of the arrival process is no longer needed in order to be able to derive the performance characteristics of the packet delay. That is, the behavior of the system contents as well as the delay may be strongly dependent on the exact nature of the arrival process, but the transformation from system contents to delay is not.
6. Application to ATM switches We have compared the results derived in this paper with simulation results, for two specific discrete-time queueing systems that can be used to model ATM switches. ATM switching networks consist of a matrix of elementary ATM switches that provide the routing of ATM cells (the equivalent of a packet in ATM) sent through the network. In the following examples, we focus attention on one specific ATM switch (Fig. 1). Cells enter the switch via one of its input links, and are then routed to one of the output links, where they are temporarily buffered in a designated output queue (O.Q.) to await transmission of the cells that arrived before them in this particular output queue and that are not yet sent. The results of this paper are applicable to assess the delay performance in such a multiple-server output queue, under the assumption that each output queue (corresponding with one destination), has a constant number of transmission lines (denoted by the integer c 2 1) via which the cells are transmitted. As shown in Fig. 1, in the following examples, N denotes the number of input and output links of a switch. Cell arrivals on different input links are assumed to be independent (since they originate from different sources), but the arrival process on one particular inlet can be either independent or correlated from slot to slot. Furthermore, it is assumed that the routing
Fig. 1. ATM switch with multiserver output queues.
Y Xiong et al. /Performance
200
1 OOE-01
-c g
Evaluation
24 (1996) 189-204
1
1
IOOE-02 -4 t
1
Fig. 2. Delay distribution
in an ATM switch with uncorrelated
arrivals on the inlets: N = 16, c = 1,2,4, p = 0.8.
of cells from the input links to the N/c output queues is performed in a uniform and independent way, i.e., each arriving cell is routed to any given output queue with probability c/N (since there are N/c possible destinations), independently of the routing of other cells. Such a model is reasonable if the cell streams on the inlets of the switch are composed of many independent multiplexed cell streams originating from users whose peak cell rate is very low in comparison with the link rate. 6.1. ATM switch with uncorrelated arrival processes on the inlets Let us first assume that the arrival process on each inlet is uncorrelated (in other words, the numbers of cell arrivals during consecutive slots are i.i.d.), and that with probability p (the load on each inlet) there will be one cell arrival during a slot on a tagged inlet, and with probability 1 -p there will be no cell arrival, that is, the number of cell arrivals during each slot on an inlet is Bernoulli distributed with parameter p. As we already mentioned, arriving cells are then routed in a uniform and independent way to the output queue assigned to their destination, which implies that the total offered load to one output queue is (T=pc. In Fig. 2, we have plotted the delay distribution d(n) versus n, for N = 16, c = 1, 2, 4, and p = 0.8. These results have been obtained through computer simulation; the solid lines connect the simulated data obtained for the delay distribution, while the data represented by the dots
I’. Xiong et al. /Performance Evaluation 24 (1996) 189-204
201
have been calculated using Eq. (19), whereby the buffer-contents distribution s(n) was also obtained through simulation. Comparison of these results gives a perfect match, which confirms the validity of the analysis presented in this paper. Note that for the model considered here, explicit expressions for the probability generating functions of the buffer contents and the packet delay are available in [14] and/or [20]. 6.2. ATM switch with correlated arrival processes on the inlets We now consider the case where the cell arrival process on each inlet of the switch is first-order Markov (note that due to the routing of arriving cells to their output queues, the cell arrival process in an output queue is no longer first-order Markov). If we define an active period on an input link as a number of consecutive slots during which a cell arrival occurs, and, similarly, a passive period as a number of consecutive slots during which there is no cell arrival on this input link, then the first-order Markov assumption means that the lengths of both active and passive periods are geometrically distributed random variables with parameters (Yand p respectively, i.e., Pr[an active period contains II slots] = (1 - ~y)~~-l,
n 2 1,
Pr[a passive period contains n slots] = (1 - j3)pnw1,
it 2 1.
I
OOE-01
Fig. 3. Delay distribution in an ATM switch with correlated c = 1,2,4, p = 0.8, K = 5.
(first-order
Markov) arrivals on the inlets: N = 16,
202
Y. Xiong et al. /Performance Evaluation 24 (1996) 189-204
Furthermore, lengths of consecutive active (or passive) periods are i.i.d. Note that the case of an uncorrelated Bernoulli arrival law on each input link of the switch corresponds to (Y+ p = 1. The couple (a, p) fully describes the arrival process on each of the inlets. Nevertheless, we prefer to use a pair of parameters that allow a more intuitive interpretation. First of all, defining p as the average load on each of the inlets of the switch, (i.e., (T=pc>, we find that p = (1 - p)/(2 - LY - p). Furthermore, we define K as the ratio of the mean length of an active (or passive) period in the correlated case versus the mean length of an active (or passive) period in the uncorrelated case with same (mean) load p. This implies that K p (1 -a)/(1 - a> = p/(1 - p). Thus, the case of an uncorrelated Bernoulli arrival law on each individual inlet corresponds to K = 1, and it is clear that the couple (p, K) also fully describes the cell arrival process on the inlets. In Fig. 3, we have plotted the delay distribution d(n) versus n, for N = 16, c = 1, 2, 4, p=O.8 and K=5. Th ese results have been obtained through simulation; again, the solid lines connect the simulated data obtained for the delay distribution, while the data represented by the dots have been calculated using Eq. (191, whereby the buffer-contents distribution was also obtained through simulation. Also for this case, comparison of these results gives a perfect match, and confirms the analysis presented in this paper, as we expected. Note that for the single-server case (c = 1) in the model considered here, the performance characteristics of the buffer contents were extensively treated in [25]. 7. Conclusions In this paper, we have established a relationship between the packet delay and the buffer contents in a discrete-time multiserver queueing system without server interruptions, where the service times of the packets are equal to one slot, and where the arrival process of the packets in the buffer is a general (not necessarily Markovian) correlated process. Specifically, explicit expressions have been derived for the distribution, the probability generating function, the mean (Little’s theorem) and the variance of the packet delay, in terms of the distribution, the probability generating function, the mean and the variance of the buffer contents. These results have been illustrated for two discrete-time queueing systems that can be used to model ATM switching elements. One of the main observations is that knowledge of the exact nature of the arrival process is not required in order to be able to derive these general relationships between the statistics of the delay and the buffer occupancy. Acknowledgement
The second author wishes to acknowledge Scientific Research (N.F.W.O.).
the support of the Belgian National Fund for
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[2] N.D. Georganas, Buffer behavior with Poisson arrivals and bulk geometric service, ZEEE Trans. Comm. 24 (1976) 938-940. [3] S.S. Lam, Delay analysis of a time division multiple access (TDMA) channel, IEEE Trans. Comm. 25 (1977) 1489-1494. [4] I. Rubin, Message delays in FDMA and TDMA communication channels, IEEE Trans. Comm. 27 (1979) 769-777. [5] H. Bruneel, A general treatment of discrete-time buffers with one randomly interrupted output line, Eur. .Z. Oper. Res. 27 (1986) 67-81. [6] H. Bruneel, Message delay in TDMA channels with contiguous output, IEEE Trans. Comm. 34 (1986) 681-684. [7] M.J. Karol, M.G. Hluchyj and S.P. Morgan, Input versus output queueing on a space-division packet switch, IEEE Trans. Comm. 35 (1987) 1347-1356. [8] A. Gravey, J.-R. Louvion and P. Boyer, On the Gee/D/l and Gee/D/l/n queues, Perf Euul. 11 (1990) 117-125. 191 E. Desmet and G.H. Petit, Performance analysis of the discrete-time multiserver queueing system GedN)/D/c/K, RACE Workshop, Munich, 3-4 July 1990. [lo] G.H. Petit and E. Desmet, Performance evaluation of shared multiserver output queue switches used in ATM, 7th ZTC Specialist Seminur, Morristown, N.J., 1990. [ll] S. Li, A general solution technique for discrete queueing analysis of multimedia traffic on ATM, IEEE Trans. Comm. 39 (1991) 1115-1132. [12] K.-T. Ko and B.R. Davis, Delay analysis for a TDMA channel with contiguous output and Poisson message arrival, IEEE Trans. Comm. 32 (1984) 707-709. [13] B. Steyaert and H. Bruneel, A general analysis of the packet delay in TDMA channels with contiguous-slot assignment, Proc. ZCC’9Z, Denver, Cola., June 1991, Vol. 3, pp. 1539-1543. [14] H. Bruneel, B. Steyaert, E. Desmet and G.H. Petit, An analytic technique for the derivation of the delay performance of ATM switches with multiserver output queues, Znt. J. Digit. & Analog Comm. Syst. 5 (1992) 193-201. [15] E. Desmet, B. Steyaert, H. Bruneel and G.H. Petit, Tail distributions of queue length and delay in discrete-time multiserver queueing models, applicable in ATM networks, Proc. ZTC’13 Copenhagen, June 1991, Vol. Queueing, Performance and Control in ATM, pp. l-6. [16] K. Laevens and H. Bruneel, Delay analysis for discrete-time queueing systems with multiple randomly interrupted servers, Eur. J. Oper. Res. 85 (1995) 161-177. [17] H. Bruneel, Packet delay and queue length for statistical multiplexers with low-speed access lines, Comput. Networks ZSDN Syst. 25 (1993) 1267-1277. [18] Y. Xiong and H. Bruneel, Buffer contents and delay for statistical multiplexers with fixed-length packet-train arrivals, Perf Euul. 17 (1993) 31-42. [19] B. Steyaert and H. Bruneel, An effective algorithm to calculate the distribution of the buffer contents and the packet delay in a multiplexer with bursty sources, Proc. GLOBECOMPI, Phoenix, AZ., December 1991, Vol. 1, pp. 471-475. [20] H. Bruneel and B.G. Kim, Discrete-time Models for Communication Systems Including ATM, Khtwer Academic Publishers, Boston (1993). [21] S.L. Brumelle, A generalization of L = AW to moments of queue length and waiting times, Oper. Res. 20 (1972) 1127-1136. [22] D.P. Heyman and S. Stidham, The relation between customer and time averages in queues, Oper. Res. 28 (1980) 983-994. [23] P.W. Glynn and W. Whitt, Extensions of the queueing relations L = AW and H = AG, Oper. Res. 37 (1989) 634-644. [24] I. Rubin, Access control disciplines for multi-access communication channels: reservation and ‘IDMA schemes, IEEE Trans. Inform. Theory 25 (1979) 516-536. [25] B. Steyaert and H. Bruneel, Analytical study of the buffer contents in an ATM switch with Markovian arrivals, Proc. ZSTN-92, Beijing, China, September 1992, pp. 9-12.
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Yiiun Xiong was born in Wuhan, P.R. China, in 1963. He received the B.S. and M.S. degrees from Shanghai Jiao-Tong University, Shanghai, P.R. China, in 1984 and 1987, respectively, and the Ph.D. degree from the University of Ghent, Gent, Belgium, in 1994, all in electrical engineering. From January 1987 to Jyly 1989 he was a Ph.D. student at the Southeast University, Nanjing, P.R. China. From August 1989 to August 1992, he was with the Research Center of Alcatel Bell Telephone in Antwerp, Belgium. Since September 1992, he has been a scientific researcher at the Laboratory for Communications Engineering, University of Ghent, Belgium. His current research interests include the modeling and performance analysis of communications systems, traffic control and management in high-speed networks, queueing theory and simulation.
Herwig Bruneel was born in Zottegem, Belgium, in 1954. He received the MS. degree in electrical engineering, the degree of Licentiate in computer science, and the Ph.D. degree in computer science in 1978, 1979 and 1984, respectively, ail from the University of Ghent, Ghent, Belgium. Since 1979, he has been working as a researcher for the Belgian National Fund for Scientific Research (N.F.W.O.) at the University of Ghent. He is also a PRofessor in the Faculty of Applied Sciences at the same university. His main researc interests include stochastic modeling of digital communications systems, , discrete-time queueing theory, and the study of ARQ protocols. He has published more than 1 70 papers on these subjects and is coauthor of the book H. Bruneel and B.G. Kim, “DiscreteTime Models for Communication Systems Including ATM” (Kluwer Academic Publishers, Boston, 1993).
Bart Steyart was born in Roeselare, Belgium, in 1964. He received the degrees of Licentiate in Physics and Licentiate in Computer Science in 1987 and 1989, respectively, all from the University of Ghent, Ghent, Belgium. Since 1990, he has been working as a Ph.D. student at the Laboratory for Communications Engineering, University of Ghent. His main research interests include performance evaluation of discrete-time queueing models.