Singles (primaries) arrive at the system at Poisson rate 
and pairs (pair = primary + secondary) arrive at the system at Poisson rate
; let
. When a primary arrives at the system, it immediately joins the M/M/1 queue, either alone or just behind a secondary that was being delayed. When a secondary arrives at the system it splits from its primary and is delayed in a separate area outside the queue. The delayed secondary is “pushed” by the next arriving primary to join the queue, followed immediately by the new primary. (Thus the delay of each secondary before joining the queue is distributed as exponential-
.) Customers in the queue are served one at a time at exponential rate 
in firstcome-first-served order. When a primary joins the queue alone, the server workload is increased by exponential-
. When a primary and secondary join the queue simultaneously, the server workload is increased by Erlang (2,
), i.e., the sum of two independent exponential-
’s. All secondaries join the queue simultaneously with (next) primaries. The number of secondaries being delayed in the system at any instant is either 0 or 1.
Define the state of the system as 
where W(t) = server workload at time
, 
if zero secondaries are being delayed, 
if one secondary is being delayed. The state with zero customers in the system is denoted by
. The state when the server is idle and one secondary is being delayed is denoted by
. Let 
denote an exponential-
random variable, and 
denote an Erlang
random variable.
Technique of “Lines and Sheets (or Pages)” We utilize a technique of “lines and sheets (or pages)” to picture the state space and a sample path in it (e.g., see Brill [
.
We select an arbitrary continuous subset in each sheet, having one boundary as a fixed level 
in the state space of
, e.g., 
,
. We use this concrete physical picture as a guide to balance the samplepath exit and entrance rates of the selected state-space subsets. Simple level-crossing theorems (e.g., Brill [
for each sheet is a unique term, or linear factor in a term, of the corresponding balance equation. The balance equations are generally Volterra integral equations of the second kind with parameter. Thus there is an isomorphism between the physical sample path structure and the model equations.
Consider the sample path of 
in 
because only primaries join the queue when arrivals find zero delayed secondaries present. All jumps due to an arrival on page 1 are distributed as 
because both the delayed secondary and the arriving primary join the queue simultaneously.
At time 0 the system is empty (state
). A single arrives and the SP (sample path) jumps to level 
on page 0; the arrival immediately starts service. The SP decreases at rate 
A pair arrives, the primary joins the queue and the secondary is delayed; the SP jumps 
since the server workload includes only those customers in the queue, and transits to page 1 (delayed secondary present).
A single arrives and pushes the delayed secondary to join the queue; the single joins just after it. The SP jumps 
and transits to page 0 (no delayed secondary present).
A single arrives and joins the queue; the SP jumps 
and remains on page 0. A pair arrives; the primary joins the queue and the split-off secondary becomes delayed. The SP jumps 
and transits to page 1. A pair arrives, the primary pushes the delayed secondary into the queue and joins just after it; the new secondary becomes delayed. The SP jumps 
and remains on page 1. The SP hits level 0 from above, and remains in state 
for a time distributed as exponentia-
(server idle). A single arrives, pushes the delayed secondary into the queue and follows just after it. The SP jumps by 
and transits to page 0. Finally the SP hits level 0 from above and enters 
(system empty).
. A cycle starting and ending in state 
would be produced similarly.