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USC Computer Science Technical Reports, no. 776 (2002)
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USC Computer Science Technical Reports, no. 776 (2002)
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Content
Instability of FIFO at Arbitrarily Low Rates in the Adversarial
Queueing Model
Rajat Bhattacharjee
Ashish Goel
University of Southern California
rbhattac, agoel @cs.usc.edu
Abstract
We study the stability of the commonly used packet forwarding protocol, FIFO (First In First Out), in
the adversarial queueing model. We prove that FIFO can become unstable, i.e., lead to unbounded buffer-
occupancies and queueing delays, at arbitrarily low injection rates. In order to demonstrate instability at
rate , we use a network of size polynomial in .
1 Introduction
In traditional queueing theory, the source which generates network traffic is typically assumed to be stochas-
tic. However, the growing complexity of network traffic makes it increasingly unrealistic to model traffic
as, say, a Poisson stream. Adversarial Queueing Theory is a robust and elegant framework developed by
Borodin et al. [5] to address this problem. In this model, packets are injected into the network by an adver-
sary rather than by a stochastic process. To keep things simple, it is assumed that the route of each packet
is given along with the packet itself. Each edge in the network can forward at most one packet in one time
step. If there are multiple packets waiting to cross the same edge, then we need a contention resolution pro-
tocol to decide which packet goes across and which packets wait in the queue. The adversary is limited in
the following way: over any window of consecutive time steps, the adversary can inject at most packets that need to traverse any edge in the network. The parameter is called the injection rate and must
be less than 1. The parameter is called the burst-size. Such an adversary is called a -adversary. Once
injected, packets follow their routes one edge at a time till they reach their destination.
Intuitively, the adversary is not allowed to introduce more traffic on an average than times the capacity
of any edge. Thus, there can be no identifiable “hot-spots” in the system. This model finds the fine middle
ground between stochastic arrivals on the one hand (as in traditional queueing theory) where packet arrival
is too predictable, and completely unconstrained adversaries on the other (as in competitive analysis), where
the adversary is allowed to over-load the system.
A packet forwarding protocol is said to be stable against a given adversary and for a given network if the
maximum queue size, as well as the maximum delay experienced by a packet, remain bounded. A packet
Department of Computer Science, University of Southern California, Los Angeles CA 90089-0781.
Department of Computer Science, University of Southern California, Los Angeles CA 90089-0781. Research supported by
NSF CAREER Award 0133968.
1
forwarding protocol is said to be stable at rate (or, -stable) if it is stable against all adversaries,
and for all networks. It is said to be universally stable if it is -stable for all . Studying the stability
of protocols was the main motivation behind the adversarial queueing theory model. In a seminal paper,
Andrews et al. [2] showed that several natural protocols are universally stable, but surprisingly, FIFO is not.
This is an important observation since FIFO is by far the most widely used scheduling protocol. It also leaves
the following question open:
Is FIFO -stable at some rate , or is FIFO unstable at arbitrarily low rates?
This is an important question, given the prominence of FIFO as the packet forwarding protocol for the Inter-
net and other networks, and has been the subject of much study over the last few years [8, 6, 7, 2, 4, 12, 19].
For several natural protocols other than FIFO, the corresponding question about the stability threshold in the
adversarial model has already been answered [22]. The problem for FIFO is particularly intriguing since
there is some intuitive evidence for each direction, due to Bramson [7, 8], arising from two somewhat unre-
lated models (more details are given in the related work section). Diaz et al. [12] improved the threshold of
instability for FIFO to ! #"%$(the original proof of Andrews et al. [2] showed instability of FIFO at rate ! #"’& )
and recently, Lotker et al. [19] further improved it to ! (& .
Our result
In this paper, we prove that FIFO can become unstable at arbitrarily low rates in the adversarial model. This
was one of the major remaining open problems in the field of adversarial queueing theory. In particular, for
an arbitrarily low injection rate , we construct a network and determine adversarial injections so that FIFO
becomes unstable. The size of our network is polynomial in )’ . This is quite strong since it even excludes
the possibility that FIFO might be stable at rate *+,-)/.10’2!3!4+ where 4 is the size of the network and 5 is a
constant 6 .
The main idea is the construction of a gadget which, assuming certain initial conditions, allows only a
small fraction of packets to pass through it for a long duration. In particular, the fraction of packets which
escape is bounded by 7!)8,/9;: where 7 is a parameter of the gadget and can be increased arbitrarily. The
network is constructed using this gadget. The adversary works in phases. At the beginning of a phase, we
assume that there are some packets waiting to pass through a column of gadgets. Using each gadget in the
column, more packets are generated which want to ultimately traverse through a second column. Additional
copies of the gadget mentioned above are used to delay and synchronize these new packets so that, at the
end of the phase, there are more packets waiting to traverse the second column than were waiting at the first
column at the beginning of the phase. Applying this inductively leads to instability. Our gadget and network
are quite different from those used in earlier works [7, 2, 22] to prove instability results.
Related Work
Andrews et al. [2] proved that rings and DAGs are universally stable networks. They showed that Longest In
System (LIS) and Shortest In System (SIS) are universally stable protocols and that FIFO is not universally
stable. In fact, they showed that FIFO can become unstable at rates greater than ! #"%& . Several natural proto-
cols such as NTG (Nearest To Go) and LIFO (Last In First Out) have been shown to be unstable at arbitrarily
low rates [22]. Goel [17] and Gamarnik [15] gave a simple and complete characterization of universally sta-
ble networks. Diaz et al. [12] improved the threshold of instability for FIFO to ! #"’$and recently, Lotker et
< It is trivial to see that FIFO (infact, any greedy protocol) is stable at rate =;>@? .
2
al. [19] further improved it to ! #& . For the case when routes are not given by the adversary, Aiello et al. [1]
and Andrews et al. [3] studied routing algorithms which ensure that no edge gets overloaded, assuming that
the adversary injects packets for which this is feasible. Gamarnik [16] showed that it is undecidable to deter-
mine whether a given protocol is universally stable, for an interesting class of protocols. Feige [13] demon-
strated non-monotonic phenomena in packet routing. Andrews [4] demonstrated the instability of FIFO in
session-oriented networks [9, 10] A .
Stability of networks has also been studied from a more queueing theoretic viewpoint in stochastic net-
works, where the packets are injected and serviced according to a stochastic process. Instability in stochastic
networks was first demonstrated by Rybko and Stolyar [21], building on the work of Lu and Kumar [20], and
Kumar and Seidman [18]. The fluid model involves taking the fluid limit of a stochastic process. Dai [11]
related stability in the fluid model to that in the stochastic model. Bramson studied the stability of FIFO in
two different stochastic models. He showed that FIFO is stable at all rates BC if packets are injected
by a Poisson process, and the time for a packet to traverse an edge is an i.i.d. exponential random variable
(i.e. the network is Kelly-type) [8]. Bramson [7] also showed that FIFO can become unstable at arbitrarily
low rates in a job-shop scheduling model if a job is allowed to visit the same machine multiple times, and is
allowed to have a different processing time on different visits to the same machine. Gamarnik [14] proved
an analogue of Dai’s result for adversarial networks.
Section 2 describes and analyzes our basic gadget. Section 3 gives the construction of a network which
is unstable at arbitrarily low injection rates. The adversarial injection patterns are described in section 4.
Finally, we prove that FIFO is unstable against this adversary in section 5.
2 The Basic Gadget
Section 2.1 describes the topology of the gadget. Section 2.2 talks about a special kind of flow. Finally, in
section 2.3 an upper bound on the number of packets that escape the gadget for this flow is proved.
2.1 The topology of the gadget
The gadget has a parameter 7 . A 7 -gadget has D’7 vertices: E 6 - - ;@E
: @
6 - - ;
: . There are four groups of
edges:
F Input edges, G 6 - - ,,G
: , pointing into E 6 - - ;@E
: respectively.
F Output edges, H 6 - - IH
: , pointing out from 6 - - ;
: respectively.
F Load edges, J 6 - K@J
: , pointing from EL to ML .
F Helper edges, N 6 - - ;@N
: , pointing from ML to ELPO
6 . The Q ’s wrap around 7 .
Note that the load and helper edges form a ring with the input and output edges pointing in and out from
alternate nodes. Figure 1 shows an example gadget.
2.2 A special flow
There are two kind of packets in this flow:
R Whether FIFO is unstable at arbitrarily low rates in the session-oriented model remains an interesting open problem.
3
helper edges load edges
output edges
input edges
...
...
...
Figure 1: A gadget
F For each Q/ST’ - - I7 , packets enter the gadget through G L at rate 1. The route of a particular packet is:
G’LK@JULV@N-L,JWLXO
6 @NLPO
6 JWLPO
A - - INLXO
:WY A JWLPO
:WY 6 H!LXO
:WY 6 Hence, each packet traverses all the load edges in the gadget. These packets are referred to as the
gadget-traversing packets.
F For each load edge, JWL , single edge packets are introduced at a rate . These packets are referred to as
the internal-gadget packets.
2.3 Upper bound on the leak
Rate of leak from a gadget, , is the sum of the rates at which the gadget-traversing packets arrive at the
source node of an output edge. In the following, we prove an upper bound for assuming the flows men-
tioned in section 2.2. For proving the upper bound we use the following property of FIFO:
Remark 2.1 Let J be an edge in the network and Z 6 ;Z
A - ;;Z\[ be ] different types of packets which ar-
rive at the source node of J at rates 6 A - - ,^[ respectively. Then the rate at which packets of type Z_L
traverse J is given by the expression, ‘baPced
^LV
L f [ gih
6 g8j
Lemma 2.2 During the time when the special flow is maintained, lk : m 6 Ooniprq
.
Proof: By symmetry, the total rate of arrival of packets at the source node of the load edges is the same. Let
that rate be .
stutu
6 v - 9
:UY 6 where WL is the rate of arrival of packets which have traversed Q of the 7 load edges. Using remark 2.1, we
obtain 6 s) . Similarly, for all DwkQxk7 ,
L sv L Y 6 )sy-)
L 4
Since vzvtu at all times, it follows that : kv)8,/ : . Therefore at all times,
{sv7!
: ku7!)8,t9 : 3 The Network Topology
The network contains three components, the columns, the connectors and the shortcuts. We describe each
of these in turn, after first describing the concept of concatenation of gadgets, which is required for each of
these components. In this section and all others, all gadgets would have the same parameter, 7 .
3.1 Concatenating gadgets
A gadget | A is said to be concatenated to a gadget | 6 if
1. The output edges of | 6 act as the load edges of | A , and
2. The load edges of | 6 act as the input edges of | A .
Note that more than one gadgets can be concatenated to a gadget, and also, a single gadget can be concate-
nated to more than one gadgets. A Chain }~sV
6 @
A - - Iw[ , of length ] , is produced by the concate-
nation of the gadget bLPO
6 to the gadget wL , for wkvQv] . A Bridge of length is said to exist between
gadgets | 6 and | A if there exists a chain | 6 6 A - - ;@\r;|
A .
3.2 Columns
The network has two separate columns, } 6 ;}
A . A column is a chain of length , where is a parameter
which will be specified later.
} 6 s} 6;6
;}
6; A - - ;;}
6; } A s} A@6
;}
A@ A - - ;;}
A@ 3.3 Connectors
There are two sets of connectors, one from } 6 to } A and the other from } A to } 6 . Connectors are bridges of
length between each gadget of a column and the first gadget of the other column, where is a parameter
which will be specified later. So, for each } 6;
L , we have the following bridge:
} 6;
LV
6;
L 6 @
6;
L A - - I
6;
L ;}
A@6
Similarly, for each } A L , we have the following bridge:
} A@
L A@
L 6 @
A@
L A - - I
A@
L ;}
6;6
5
C1
Column
C2
Column
Figure 2: Topology of the network: Shortcuts are not shown
3.4 Shortcuts
Shortcuts are bridges of length from each connector gadget, 6;
L g to } 6;6
. We refer to the respective short-
cut gadgets as 6;
L g . Similar shortcuts exist between the other set of connector gadgets and } A@6
.
Figure 2 shows the schematic of the network topology.
4 The Adversarial Injection Pattern
The adversary introduces two kinds of flows of packets (in addition to the internal gadget packets). Flow
through the columns and flow through the connectors. The description of both assumes the concept of acti-
vation of a gadget and sequential activation of gadgets in a chain, which are described next. We will assume
that there is a chain | 6 I|
A - - ,;|- . These gadgets will be activated one after the other.
A packet is considered to be a chain-traversing packet for a chain | 6 I|
A - - ;I| if the route of the
packet is gadget-traversing for each |L in the chain.
Figure 3: Route of a chain-traversing packet
6
4.1 Activation of a gadget
Precondition: There are gadget-traversing packets in the queue of each input edge of the gadget |\L . Also,
there is no other packet in any other gadget in the chain. Moreover, the packets are chain-traversing for the
chain |LV;|LPO
6 - - ;I| (we will slightly modify this condition later).
Activation: During the activation phase, the adversary introduces internal-gadget packets at rate in the
gadget |L . The activation phase lasts for time steps. Note that during this phase the internal-gadget packets
and the gadget-traversing packets compete with each other to traverse the load edges of the gadget.
Postcondition: There are K packets in the queue of each load edge of the gadget |L and there are
no other packets in any other gadget. Each packet is chain-traversing for the chain |wLXO
6 ;|LXO
A - - ;;| .
Later a lower bound for the quantity will be presented. In order to ensure that these conditions are met,
we modify the injection in the following way (we essentially use the fact that the routes of all the packets
are predetermined): (1) In order to get rid of internal-gadget packets which might be interleaved with the
chain-traversing packets in the queue (after time steps), we make sure that these particular internal-gadget
packets are never introduced. (2) To avoid the case when packets in the queue of the load edges have other
edges, within the gadget |L , to traverse, we modify the routes of the packets such that after time steps, they
are simply queued in one of the load edges of |L and the next edge on their route is a load edge of gadget
|LXO
6 . This can be done since it amounts to merely taking away some of the edges from the path of the packets
and can only reduce the injection rate. (3) We will assume that the routes of all the packets which escaped
the gadget during the time steps end at gadget |L itself. This ensures that there are no packets in the other
gadgets.
Sequential activation of gadgets: In a chain of gadgets, postcondition of the activation of a gadget acts
as the precondition for the activation of the concatenated gadget. Sequential activation of gadgets is the cas-
cading activation of gadgets starting from some initial chain-traversing packets waiting in the queue of the
input edges of the first gadget.
4.2 Flow through the columns
Time steps are grouped into phases. It is assumed that at the beginning of an even phase there are chain-
traversing (for column } 6 ) packets in the queue of each of the input edges of the gadget } 6;6
. We show that
at the end of the phase there are more than chain-traversing (for column } A ) packets waiting on each of
the input edges of the gadget, } A@6
. Similarly, at the beginning of an odd phase, there are chain-traversing
packets in the queue of each of the input edges of the gadget, } A@6
and at the end of the phase there are more
than chain-traversing packets in the queue of each of the input edges of the gadget, } 6;6
. We only show
packet injections for the even phases, packet injections for the odd phases are similar. Applying the above
repeatedly leads to instability.
Subphases: Each phase is divided into subphases. During the QVr subphase the gadget } 6I
L is activated.
7
4.3 Flow through the connectors
At the end of subphase Q , let there be WL packets in the queue of each of the input edges of } 6;
LXO
6 . During
the next WL time steps (i.e. the subphase Q ), WL)’7 packets each are introduced in the queue of each in-
put edge of } 6;
LXO
6 . The route of these packets are chosen such that they are chain-traversing for the chain
V
6I
L 6 6;
L A - ;
6I
L g 6;
L g ;}
A@6
I}
A@ A - - I;}
A@ , for some 9 . After L time steps, these packets
form the precondition for the sequential activation of the gadgets mentioned above. This is because there are
UL packets present in the queue of each input edge of } 6;
L ahead of the newly injected packets. The routes of
the packets are such that at the end of the phase i.e. at time
f L h 6 L , they are all queued at the input edges of
the the gadget, } A@6
. This can be achieved using the shortcut gadgets and the fact that the route of each packet
is determined in advance ( can be chosen arbitrarily). Observe that the routes of the chain-traversing
packets and the internal-gadget packets introduced during a subphase are mutually exclusive. Moreover, the
injection rate of each type of packet is less than for any edge.
5 Proof of Instability
This section shows that at the end of a phase, there are more packets queued at the input edges of } A@6
, than
were queued at the input edges of } 6;6
. We will ignore floors and ceilings, since losses caused due to them
can be offset by assuming that is large enough. To initially generate constant number of packets to start
phase 0, we can attach large acyclic graphs to the input edges of } 6;6
, where the acyclic portion is used to
generate the initial packets.
Analogous to our description of the flow, the analysis can be broken down into two parts: flow through
the columns and flow through the connectors.
5.1 Flow through the columns
First we set the values of the various parameters. Let 7 be such that ,^v : $%D’7¢¡)% A , s£¢7!)’ ,
⁄sy-¥’7 A )’ A .
Lemma 5.1 Let L be the duration for which the gadget } 6;
L remains activated during the QVƒ subphase. For
kQ§k , the number of packets in the queue of each of the input edges of the gadget } 6;
L at time
f L Y 6 g;h
6 g ,
i.e. the beginning of the Q ƒ subphase, is lower bounded by -)%D .
Proof: Recall that the duration for which a gadget remains activated is the same as the number of pack-
ets waiting at each input edge when the gadget gets activated. is also the number of packets waiting at the
Observe that 6 s . Lemma 2.2 implies that for a gadget activated for -L time steps, the total number of
packets which leak through in the L steps is at most
UL
7 ,t9 : By our choice of and 7 , it follows that : m 6 Oon;p q v-)%D . We now have
UL¤z UL
Y 6 ' «“
7 i‹u :¢›
8
z ' «“
7 ,/ :¢›
[since Q§k ]
z ' «“
7 i‹u : › z ,«“9)’D’
s -)’D
Hence, the lemma follows.
5.2 Flow through the connectors
Recall that for each kfiQfl , a set of WL)’7 (for each edge) chain-traversing packets for the chain:
} 6I
L 6I
L 6 6;
L A - - ;I}
A@6
- ;;}
A is introduced during the subphase Q8 .
Lemma 5.2 After the packets have activated ^“ gadgets in a connector, the number of packets queued at
each input edge of the connector gadget 6;
L g is lower bounded by -)£7 .
Proof: Using arguments similar to the one used in Lemma 5.1, we can conclude that the number of packets
queued at each input edge of the gadget 6;
L g (after w“9 gadgets have been activated) is at least:
UL
7 ' «“
7 it9 : › By the choice of and 7 , : m 6 Oon;p q v-)’D and since WL‹u-)’D , the number of packets is at least:
D%7
' «“
7 ,§9– :!›
z D%7
,“-)’D%†s
–
£¢7
Hence, the lemma follows.
Lemma 5.3 At time
f L h 6 L , i.e. at the end of all subphases, the number of packets queued at the input edges
of } A@6
is greater than .
Proof: We first show that is large enough to allow the different sets of chain-traversing packets injected
during each of the subphases to simultaneously arrive at the input edges of } A@6
. This can be accomplished
if the time step at which 6;
L (for all kQ§k ) is activated is greater than
f L . By Lemma 5.2, the time
step at which 6;
L is activated is at least:
£¢7
⁄s
£7 A sv‡z· UL
Using Lemma 5.2, the number of packets on the queue of each input edge is at least:
‡-)£7wsv
Hence the lemma.
We now state the main theorem of our paper.
9
Theorem 5.4 FIFO is unstable for arbitrarily low injection rates.
Proof: In light of Lemma 5.3, we only need to show that the rate of injection is always less than . The
latter is true because the gadget-traversing packets and the internal-gadget packets are always injected for
mutually exclusive set of edges and these packets themselves respect the injection rate (See section 4 for
details).
We have ignored floors and ceilings in this proof. Losses caused due to floors and ceilings depend only
on the size of the network and , and not on . These losses do not matter if is large enough; we omit the
details from this version.
5.3 The size of the network
We now show that the size of the network is polynomial in -)’ . Let 7 be 5 6 n .10’2 6 n for a large enough 5 .
,/ : uD’3U¶;•
m < ‚ p zu$’D’7 ¡ )’ A Therefore, 7ws*+ 6 n .10’2 6 n ; . Now, the size of the network is *+‡7!/s*„7!”)’ ¡ , which is polynomial in
-)’ . This is quite strong since it even excludes the possibility that FIFO might be stable at rate *„,-)x.10’2 3 4+
where 4 is the size of the network and 5 is a constant.
10
References
[1] W. Aiello, E. Kushilevitx, R. Ostrovsky, and A. Rosen. Adaptive packet routing for bursty adversarial
traffic. Proceedings of the 30th Annual ACM Symposium on Theory of Computing, pages 359–368,
1998.
[2] M. Andrews, B. Awerbuch, A. Fernandez, J. Kleinberg, T. Leighton, and Z. Liu. Universal stability
results for greedy contention-resolution protocols. 37th IEEE symposium on Foundations of Computer
Science, pages 380–389, 1996.
[3] M. Andrews, A. Fernandez, A. Goel, and L. Zhang. Source routing and scheduling in packet networks.
42nd IEEE Symposium on Foundations of Computer Science, 2001.
[4] Matthew Andrews. Instability of fifo in session-oriented networks. Proceedings of the eleventh annual
ACM-SIAM symposium on Discrete algorithms, pages 440–447, 2000.
[5] A. Borodin, J. Kleinberg, P. Raghavan, A. Sudan, and D. Williamson. Adversarial queueing theory.
28th ACM Symposium on Theory of Computing (Also, to appear in JACM), pages 376–385, 1996.
[6] M. Bramson. Instabilityof FIFO queueing networks. Annals of Applied Probability, 4(2):414–31, 1994.
[7] M. Bramson. Instability of FIFO queueing networks with quick service times. Annals of Applied Prob-
ability, 4(3):693–718, 1994.
[8] M. Bramson. Convergence to equilibria for fluid models of FIFO queueing networks. Queueing Sys-
tems: Theory and Applications, 22:5–45, 1996.
[9] R. L. Cruz. A calculus for network delay, part i: Network elements in isolation. IEEE Transactions on
Information Theory, pages 37:114–131, 1991.
[10] R. L. Cruz. A calculus for network delay, part ii: Network analysis. IEEE Transactions on Information
Theory, pages 37:132–141, 1991.
[11] J.G. Dai. Stability of open multiclass queueing networks via fluid models. In F. Kelly and R.J. Williams,
editors, Stochastic Networks, volume 71 of The IMA volumes in mathematics and its applications, pages
71–90. Springer-Verlag, 1995.
[12] Josep Diaz, Dimitrios Koukopoulos, Sotiris E. Nikoletseas, Maria J. Serna, Paul G. Spirakis, and Dim-
itrios M. Thilikos. Stability and non-stability of the fifo protocol. Proceedings of the Thirteenth Annual
ACM Symposium on Parallel Algorithms and Architectures, pages 48–52, 2001.
[13] U. Feige. Nonmonotonic phenomena in packet routing. STOC, pages 583–591, 1999.
[14] D. Gamarnik. Stability of adversarial queues via fluid models. 39th IEEE Symposium on Foundations
of Computer Science, 1998.
[15] D. Gamarnik. Stability of adaptive and non-adaptive packet routing policies in adversarial queueing
networks. Proceedings of the 31st ACM Symposium on Theory of Computing, 1999.
[16] D. Gamarnik. On deciding stability of some scheduling policies in queueing systems. SODA, 2000.
[17] A. Goel. Stability of networks and protocols in the adversarial queueing model for packet routing.
Networks, 37(4):219–224, 2001.
i
[18] P.R. Kumar and T. Seidman. Dynamic instabilities and stabilization methods in distributed real-time
scheduling of manufacturing systems. IEEE Transactions on Automatic Control, AC-35:289–298,
1990.
[19] Zvi Lotker, Boaz Patt-Shamir, and Adi Rosen. New stability results for adversarial queuing. Pro-
ceedings of the Fourteenth Annual ACM Symposium on Parallel Algorithms and Architectures, pages
192–199, 2002.
[20] S.H. Lu and P.R. Kumar. Distributed scheduling based on due dates and buffer policies. IEEE Trans-
actions on Automatic Control, 36:1406–1416, 1991.
[21] A.N. Rybko and A.L. Stolyar. Ergodicity of stochastic processes describing the operation of open
queueing networks. Problems of Information Transmission, 28:199–220, 1992.
[22] T. Tsaparas. Stability in adversarial queuing theory. M.Sc Thesis, Department of Computer Science,
University of Toronto, 1997.
ii
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USC Computer Science Technical Reports, no. 779 (2002)
Description
Rajat Bhattacharjee, Ashish Goel. "Instability of FIFO at arbitrarily low rates in the adversarial queueing model." Computer Science Technical Reports (Los Angeles, California, USA: University of Southern California. Department of Computer Science) no. 776 (2002).
Asset Metadata
Creator
Bhattacharjee, Rajat
(author),
Goel, Ashish
(author)
Core Title
USC Computer Science Technical Reports, no. 776 (2002)
Alternative Title
Instability of FIFO at arbitrarily low rates in the adversarial queueing model (
title
)
Publisher
Department of Computer Science,USC Viterbi School of Engineering, University of Southern California, 3650 McClintock Avenue, Los Angeles, California, 90089, USA
(publisher)
Tag
OAI-PMH Harvest
Format
12 pages
(extent),
technical reports
(aat)
Language
English
Unique identifier
UC16269699
Identifier
02-776 Instability of FIFO at Arbitrarily Low Rates in the Adversarial Queueing Model (filename)
Legacy Identifier
usc-cstr-02-776
Format
12 pages (extent),technical reports (aat)
Rights
Department of Computer Science (University of Southern California) and the author(s).
Internet Media Type
application/pdf
Copyright
In copyright - Non-commercial use permitted (https://rightsstatements.org/vocab/InC-NC/1.0/
Source
20180426-rozan-cstechreports-shoaf
(batch),
Computer Science Technical Report Archive
(collection),
University of Southern California. Department of Computer Science. Technical Reports
(series)
Access Conditions
The author(s) retain rights to their work according to U.S. copyright law. Electronic access is being provided by the USC Libraries, but does not grant the reader permission to use the work if the desired use is covered by copyright. It is the author, as rights holder, who must provide use permission if such use is covered by copyright.
Repository Name
USC Viterbi School of Engineering Department of Computer Science
Repository Location
Department of Computer Science. USC Viterbi School of Engineering. Los Angeles\, CA\, 90089
Repository Email
csdept@usc.edu
Inherited Values
Title
Computer Science Technical Report Archive
Coverage Temporal
1991/2017
Repository Email
csdept@usc.edu
Repository Name
USC Viterbi School of Engineering Department of Computer Science
Repository Location
Department of Computer Science. USC Viterbi School of Engineering. Los Angeles\, CA\, 90089
Publisher
Department of Computer Science,USC Viterbi School of Engineering, University of Southern California, 3650 McClintock Avenue, Los Angeles, California, 90089, USA
(publisher)
Copyright
In copyright - Non-commercial use permitted (https://rightsstatements.org/vocab/InC-NC/1.0/